Is The Climate Chaotic?

Guest Post by Willis Eschenbach

After I’d published my previous post on the Hurst Exponent entitled A Way To Calculate Effective N, I got an email from Dan Hughes which contained a most interesting idea. He thought that it would be productive to compare the Hurst analysis of the records of weather phenomena such as temperatures and the like, with the Hurst analysis of the corresponding climate model outputs. He proposed that we take a look at the question and discuss and share our findings. Hey, what’s not to like?

Along the way, as such things happen, the topic of our discussion and investigation turned to a larger and more interesting question—`which of the various natural datasets (temperatures, rainfall, eruptions, pressures, etc) and/or the corresponding global climate model outputs are chaotic?

I must, of course, start with the obvious question … what is meant by “chaotic”? A chaotic system is a system wherein nearby initial states either converge or diverge exponentially. An example is the ocean surface. If you drop two sealed empty bottles overboard in mid-ocean, one on each side of a boat, they will drift apart over time. This separation will be slow at first, and then faster and faster as the two bottles encounter different winds and currents in different areas.

Whether a dataset is chaotic or not is generally assessed by looking at the Lyapunov exponent. This is a measure of the “stretching factor”. In our ocean example above, the stretching factor measures how fast the two points are moving apart over time. In a chaotic dataset, the stretching factor generally increases or decreases with time. In non-chaotic datasets, on the other hand, the stretching factor doesn’t vary with time. Figure 1 shows the “Lyapunov curves” of the evolution of the stretching factor with time for a variety of natural and calculated datasets.

lyapunov curves variousFigure 1. Lyapunov curves for a variety of datasets. All datasets have been detrended and standardized before analysis.

Now, I’ve graphed four types of datasets above, indicated by the four colors. The first type, shown in red and mostly obscured by the blue lines, shows four different varieties of random numbers—normal, uniform, poisson, and high Hurst exponent fractional Gaussian random numbers. Basically the Lyapunov curves of the random number datasets are all plotting right on top of each other. Starting from time = 0, they climb rapidly to their maximum value and then just stay there. As we would expect from random data, there’s no trend in the stretching factor over time.

The next group, in blue, shows the Lyapunov curves for a half-dozen climate-related datasets, viz:

• HadCRUT4 Monthly Mean Surface Air Temperatures 1850-2015

• Annual Nilometer Minimum River Height 622-1284

• Stockholm Monthly Tides 1801-2001

• Central England Temperature Record Daily Maximum Temperatures 1878-2015

• Armagh Ireland Daily Mean Temperatures 1865-2001

• Annual Average Nile Flow, Cubic Metres/Sec 1870-1944

As you can see, in terms of the Lyapunov analysis, all six of these climate-related datasets (blue lines) are indistinguishable from the four random datasets (red lines), which in turn are indistinguishable from each other. None of them show any trace of chaotic behavior.

Another group of datasets, those at the bottom in gold colors, are quite different from the random and the observational datasets. They are a variety of chaotic datasets. Note that they all share a common factor—as mentioned above, over time the rate of separation (as measured by the “stretching factor”) increases. The rate of separation doesn’t just go to a certain high point and stay there like the random or climate-related datasets did. The rate of separation in chaotic datasets continues to rise over time.

Finally, there are a couple of other datasets in purple. These show observations of phenomena that are usually thought of as “forcings”. One is changes in the solar activity, with daily sunspots as a proxy for the activity, from 1880 to 2015. The other is annual aerosol optical depth from 800 to 2000, which is generally a function of volcanic action and is calculated from ice core data. Curiously, these two datasets plot somewhere in between the random observations at the top, and the chaotic datasets at the bottom. In addition, both of them show significant variation in stretching factor over time. The sunspots have a slight but significant increase. The aerosol optical depth goes down and then looks like it’s starting back up. So it seems that these two datasets are weakly chaotic.

Now, these results were a great surprise to me. I’ve long believed, without ever checking it, that the climate was chaotic … however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest. Hey, what do I know … I was born yesterday.

Are there any climate datasets which are chaotic, even weakly chaotic? I think so. It appears that tropical ocean temperatures are weakly chaotic … but that’s a question for the next post, which will look into Dan Hughes’s idea regarding the use of the Hurst analysis to distinguish between chaotic and non-chaotic datasets.

Regards to everyone,

w.

MY USUAL REQUEST: If you disagree with someone, please quote the exact words you disagree with. That way, we can all understand the exact nature of your objection.

CODE: To calculate the Lyapunov exponent I’ve used the lyap_k function from the R package tseriesChaos. Here are the functions I used to make Figure 1:

lineout=function(col="black",testdata=testdata,line0=F,lwd=3,s=110){

  output <-lyap_k(scale(dtrendlin(testdata)), m=3, d=2, s=s, t=40, ref=100, k=4, eps=4)

  lines(output,main="",cex.main=.95,col=col,lwd=lwd)

  invisible(output)

}

plotout=function(col="black",testdata=testdata,line0=T,lwd=3,s=110){

  output <-lyap_k(scale(dtrendlin(testdata)), m=3, d=2, s=s, t=40, ref=100, k=4, eps=4)

  plot(output,main="",cex.main=.95,new=F,ylim=c(-6,1),col=col,lwd=lwd,

       ylab="Log(Stretching Factor)")

  if (line0) abline(h=c(0,1),lty="dotted")

  invisible(output)

}

dtrendlin=function(timeseries,doplot=FALSE){

  outseries=timeseries

  thelm=lm(timeseries~time(timeseries))

  thegood=which(is.finite(timeseries))

  outseries[thegood]=outseries[thegood]-thelm$fit

  if (doplot){

    plot(outseries)

  }

  outseries

}
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237 thoughts on “Is The Climate Chaotic?

  1. Very interesting. I’ve always felt that the true chaos of climate comes from the interaction of the various systems – that is, taken in isolation, I can say that A will lead to B. But when A and B also interact with C, all bets are off. Looking forward to your future posts.

  2. The solution of the 3-body problem is chaotic but the position of the centre of gravity is not. Looking at smoothed curves could well mask the chaotic nature of the underlying system.

    • Is this true? Can you elaborate? Or provide a reference?
      It seems to me that location of the center of mass is a simple function of the location of the three body masses; these latter three locations are chaotic by definition. How can one prove the “average” of these three chaotic paths is a non-chaotic path?

      • Apply the law of conservation of momentum as the centre of mass of the 3 particles is not acted on by any external force to the 3-body system hence its motion continues unchanged.

    • I’m more inclined to think that you are looking at too short a time period. Try a period that includes several ice ages. I am beginning to think that the climate is a chaotic system with two stable points: temperate and ice age.

      • I believe the chaotic system has three attractors: Warm (dinasours, huge ferns, lots of water), Temperate (anything from the Roman warm period to the Maunder minimum) and Cold (ice age). I’d rather have warmer than colder.

    • Are we not living inside a thermodynamic heat engine? I think the answer is a big YES! And the climate is a physical & observable representation of entropy.

  3. Could you do the same for the Greenland GISP ice core for the holocene (and sub-periods therein)? Would be interesting to see what behavior there is on longer timescales.

    • Could you – or someone – do the same for the Greenland GISP ice core for the Holocene (and sub-periods therein)?
      Fixed.
      W. is my age (ish, OK).
      Spread the load.
      Have a grand week-end.
      Auto

  4. A few thoughts/questions.
    A system doesn’t have to be chaotic to be unpredictable.
    Weather does seem to be truly chaotic (within certain boundaries).
    Can a system with multiple co-dependent positive and negative feedbacks NOT be chaotic?

    • Any ‘system’ (needing a lot more precise definition) with the following two attributes is by definition mathematically chaotic. But that statement still does not define the time scales on which chaos becomes evidenced.
      1. Feedbacks. These make the system nonlinear.
      2. Feedback time lags (feedbacks do not operate instanstaneously). That makes the feedbacks time dependent.
      It is time dependent feedbacks that define chaotic systems.
      Now, with respect to weather, feedbacks are self evident. Willis tropical Tstorm thermoregulator is but one example. But we can have rip-roaring good arguments about what ‘time dependent’ means in a climate/weather context. An instant? An hour? A single diurnal cycle…? For my peer reviewed paper showing chaos in a large assembly plant, the least time interval turned out to be one shift. Anything less was ‘instantaneous’ and did not result in chaos. The ‘system definition’ matters.

      • 1. Feedbacks. These make the system nonlinear.
        Feedback can be a very linear thing. You have confused something here. In order for a system to be chaotic, it must have feed back (sort of). Just because it has feedback doesn’t mean it is either nonlinear or chaotic.
        2. Feedback time lags (feedbacks do not operate instanstaneously). That makes the feedbacks time dependent.
        Nope. TIme lags are not required, per se. This is a swing and a miss. Many systems have time lags and are not chaotic.
        This is why I think Willis does WUWT a disservice. Because he encourages these sorts of posts. The poor guy who asked the question, “Can a system with multiple co-dependent positive and negative feedbacks NOT be chaotic?” and gets a BS response. He is baffled by the BS, because it’s BS, and has to deal with figuring it out for himself.

    • “Can a system with multiple co-dependent positive and negative feedbacks NOT be chaotic?”
      It depends on what you mean by co-dependent. It is entirely possible either way. To investigate your thinking, write out your transfer function and see if there are any “cross terms”.

  5. Excellent thread, Willis. I’m looking forward to further analysis of this by you and Dan. On other threads, in response to the inimitable Dr. Robert Brown, I believe I have been the only one, or one of few that have taken issue with the dismissive idea that climate is essentially chaotic. My view comes NOT from mathematical analysis, which I am pleased is being done here, but from the relatively few different types of days (10?) I’ve experienced over almost 80yrs (~30,000days). Weather has a small deck of cards that it deals out and the conditions that give rise to these days are not much of a surprise. In eastern Ontario, we know what we are in for with a strong northwest wind in fall, winter and spring and southerlies from the Gulf of Mexico or the wet nor’easters that we get the tail end of from the Atlantic.
    Yes, climate is a complex subject with coupling of all its parts and physical states. And yes, at the molecular level it must be chaotic. And yes, weather forecasts ‘deteriorate’ by the end of a week. But what follows this forecast, I’ve probably seen 3000 times. I think this gives an inkling of a possibility in the future for more deterministic forecasting.

    • “at the molecular level it must be chaotic”
      No. At the molecular level it is random. Random is not chaotic. This is the fundamental area where Wiliis has failed. He does not understand the difference between random and chaotic, hence he conflates the two and then confuses people.

      • “No. At the molecular level it is random. Random is not chaotic.”
        Disagree. What appears at first blush to be random, if examined, will reveal a lower level of order. Examining that order will find at its root what appear to be one or more chaotic elements, which upon close examination will actually be underlain by lower level order. Whether this continues ad infinitum I don’t know.
        At best we can say that things we do not understand appear to be random. The behaviour (future states) of chaotic systems are hard to predict, not unpredictable.

      • Like the two bottles released in the ocean that drift apart, if at some instance we could clone a parallel universe to ours in which the only difference was an extra flap of a butterfly wing anywhere in the parallel Earth, then we could observe that the weather systems between the two Earths would eventually diverge.

      • You can disagree as much as you want but Einstein did some of his best work deriving from first principles the equations explaining Brownian motion. Go ahead. Find his error. We are all waiting. There’s a Nobel prize waiting for you at the end of your investigation.

      • Dinostratus
        October 23, 2015 at 11:14 pm
        “No. At the molecular level it is random.”
        Well in crystallization of water and nearly all elements of the universe, gather themselves together in wonderful order, each creating crystals of a class that we can predict for them in future crystal growth based on basic “stacking” geometries. When a granite crystallizes, the details are remarkable. Potash and sodium ions will pass each other in the liquid on their way to attach to aluminum silicon and oxygen to form two different feldspars. The provenance of unattached individual species and the level of crystallization of the melt results in “shortages” of certain elements for the simple feldspars and crystallization of different alkali-alumino-silicates species perforce organize themselves and some substitution occurs in given species as a “second” choice
        You have to define more narrowly the randomness in this ordered picture. You are likely right that there is no way to predict which individual molecules of a population will form part of a given crystal ordered arrangement, although I would suggest that the nearest ones to the crystallization site has the best chance to be ‘selected. You random guys play around too much with a mix of colored balls in a jar in your understanding of what is random.
        Let me offer, perhaps, a new idea: random itself means infinitely chaotic!!!
        Crispin in Waterloo but really in Beijing
        October 24, 2015 at 7:28 am Yes, you get it
        noaaprogrammer
        October 24, 2015 at 8:45 am

  6. Well that surprised me.
    I’ve always assumed that weather and climate are chaotic.
    Weather is still the epitome of a chaotic system, right? So how can the long-term integral of weather not be chaotic?
    This confuses me.

    • One of the standard illustrations of chaos theory is the “Butterfly Effect”. In a short timeframe weather can still be chaotic, but this suggests that in the long term all the butterflies in China can’t melt a glacier in Greenland.

      • MCourtney, all chaotic systems are still bounded by their strange attractors in N-1 Poincare space. Buzz Lightyear’s “to infinity and beyond” does not happen.
        Local weather may well be chaotic, but as one averages over larger regions and time frames this can get ‘buried’ in non-chaotic macrophenomena. As an example, Feigenbaum’s thesis on chaotic bifurcations looked at dripping water faucets. Regular drips, then a different pace of regular drips (quickening, a bifurcation event) then erratic drips (chaos), then back to slow steady drips. Transitions through 3 attractors. BUT, if you integrate the volume of water lost over suffcient time, guess what, a predicable non-chaotic value for the leak per unit time (say a day). Defining the system is an important part of saying what might be technically chaotic. Individual drip timing, yes; water loss over sufficient time, no. Mocroturbulence and drag over an airplane wing, yes since governed by Navier Stokes; calculation of average lift for the wing at some speed, no. Or we would not have flying machines. Same idea.
        Since climate is the average of weather over at least 30 years, it is quite possible observed ‘bulk’ climate is not chaotic in the mathematical sense, while still possible that the climate models which simulate it are.

      • I think you need much more evidence to convince me that 1. weather being chaotic, 2. weather being an instance of climate, the climate would not be chaotic simply due to chaotic being self-similar at multiple scales. You are like Willis and most people confusing Chaotic a mathematical quality with erratic behavior a physical quality; some very chaotic system have long periods of periodic behavior. As far as sensitive dependence on initial conditions go, it’s not about how many butterflies as much as it’s about the correct butterfly.

      • Yes. And that is what ristvan implies.
        That the weather has a strange attractor that makes temperature, air pressure, cloud cover (etc.) consistent in the long term.
        Or the medium term as it must have at least two strange attractors in the long term to create both now and ice ages.
        But… how the, what the, Wow!
        That is weird.
        We still have weather.

      • Not necessarily stable, but predictable. Could be natural oscillation, like apparently Arctic ice. Essay Northwest Passage, in part. All still bounded given whatever time frame.

  7. The function lyap_k estimates the largest Lyapunov exponent of a given scalar time series using the algorithm of Kantz. This implies that your plots are of the MLE (maximum Lyapunov exponent). The normal interpretation of the MLE for a time series is as follows: Negative Lyapunov exponents are characteristic of dissipative or non-conservative systems; A Lyapunov exponent of zero indicates that the system is in some sort of steady state mode; A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). How did you arrive at your interpretation of the results?

  8. Willis, I know nothing of Lyapunov curves. It would be helpful if you could give a layman’s definition. However, it seems to me that all climate variables are chaotic, within the bounds of the climate in question. Example: for over the past century years, temperature in Los Angeles in January never exceeded 95°F and never fell below 28°F. But nobody can predict, within a degree or two, what the high and low temperature will be next January.
    According to Wikipedia chaos is: When the present determines the future, but the approximate present does not approximately determine the future. Systems are chaotic if very small difference in initial conditions yield widely different outcomes. It is this feature that make climate hard to predict and even harder to model. If you expanded your curves so we could better see the ups and downs, we should see chaos, as we see in the aerosol optical depth curves. Optical depth is greatly influenced by volcanic action. I don’t think anyone would claim that volcanos occur in a regular and predictable manner.

  9. I am not skilled in R so cannot be precise, but something here does not make general sense. The Maximal Lyapunov Exponent (there are many Lyapunovs in a complex phase space, the maximal one being diagnostic because it eventually swamps all lesser Lyapunovs–just a way to quantify sensitive dependence on inital comditions, the ‘butterfly effect’) is diagnostic of chaos iff it is positive. Figure 1 shows several presumably ‘chaotic’ datasets (deep yellows) with negative Lyapunov exponents. Ordinarily, negative Lyapunov exponents are indicative of non-chaotic damped dissipative oscillatory systems, like a weight on a spring or a free swinging pendulum (the classic first year physics examples). And, the random data (obviously not a chaotic system) has a positive Y value when it arguably should be zero. And, the Lyapunov value is normally extracted from a time series as a single number over entire series extent, not a value that changes with length of time series. Perhaps Fig 1 y axis is not the MLE? But then it is not a mathematically proven diagnostic of true chaos.
    Ordinarly I would like to be more helpful than to merely point out something that appears out of kilter. No time for more study this week, and no desire to learn R to maybe understand what might be off.

    • I saw a definition of a chaotic system. I do not know what a chaotic dataset is. Is a sequence of random numbers chaotic? Can you have a two-dimensional chaotic dataset? Where can I find a definition of a “chaotic dataset”?

  10. I have little knowledge about chaos theory, and I’m probably not using the right wording, but what if the climate variables are /constrained/ chaotic? A bit like having a bunch of elastic balls that are bouncing in a box? The position of each ball over time is chaotic (highly dependent of initial position), but none of the balls ever make it outside the box.

    • FJ, Long ago I wrote a paper applying chaos theory to microeconomics (productivity) with a real world factory example. All chaotic systems are still bounded by their strange attractors. Those constrain any chaotic system. See my comments above for further clarification.

  11. Yes, my thoughts on chaos in weather have been modified by recent understanding of SSW (sudden stratospheric warming). That is, our problem has been ignorance. Probably we still don’t know the whole of it, and it only seems chaotic.

    • Brett
      Until a few years ago people were not aware of giant waves that could swallow ships. Similarly you can add heat pulses to the panoply of things we really know nothing about. A heat pulse caused an unexpected temperature of 164 degrees f a few years ago in Portugal
      There are lots of things we do not yet really know much about
      Tonyb

      • Tony, rogue waves only seem chaotic because rare. They are just the interference patterns of two non-parallel otherwise well behaved non-chaotic oscillatory wave patterns. In that sense, they are predictable (in theory) and mathematically non-chaotic.
        BTW rogue waves are truly scary. Happen all the time on the Great Lakes because of the bathtub effect (regular waves hit shore at an angle, reflect at the opposite angle, then interfere up to 5 miles offshore!). Took my 36 foot sailboat through some Lake Michigan 15 footers a day I hope never to repeat. Three feet of blue water over the battened topsides and into my chest behind the wheel aft. Broke the back of the Edmund Fitzgerald on the east end of Lake Superior; Famous song by Gordon Lightfoot.

  12. Um. My mother already knew the climate was chaotic 50 years ago. You never know when you’re going to need your jumper, or an umbrella.

  13. This was done a decade or two ago, and if my memory serves me correctly, the comparison was not too favorable at that time. I will try to find the paper. I can’t even recall the journal it was in.

    • Thanks, Kevin. I’d be interested in seeing whatever you have regarding the use of hurst exponents. I’m using the method of Koutsoyiannis, so my view will be different, but I’m always interested in prior art.
      w.

  14. It’s quite a while since I worked on chaotic systems, but I’d say that your quest to characterize the weather/climate via Lyapunov exponents is doomed. Typical chaotic systems can be described by just a few dynamical variables, certainly less than 10, e.g., the Lorenz equations which only have three variables. When there are a large number of dynamical variables it’s more realistic to simply characterize the system as turbulent or random.
    A simple example is provided by experiments done on Rayleigh-Benard convection where a fluid cell is heated from below and held at constant temperature at the top. When the heating is small there is no convection. Once the bottom plate is warm enough simple periodic convection cells develop. As the heating is increased the cells become more complex and the period doubles. A bit more heating results in more complexity and another doubling of the period. Eventually after just a slight bit more heating the motion becomes chaotic and can still be described as a low dimensional system with Lyapunov exponents. However, a little bit more heating and it becomes turbulent and you can no longer characterize the fluid motion by a chaotic attractor and Lyapunov exponents. In principle, solving the Navier-Stokes equations of fluid dynamics would explain all this. In practice, meh. Compare and contrast to the weather, which is also supposed to be described by the NS equations.
    If you do compute the Lyapunov exponents it’s very important to sample the time series at the correct rate, just as with data that is going to be used for a Fourier transform. A chaotic systems has to be sampled at three times the rate of the longest period in the signal rather than twice the period if a Fourier series is what you want. You are also assuming that the time series that you are sampling actually contains information from all the underlying dynamical variables. This is a stretch when you don’t understand the system, i.e., the weather.
    A more fruitful avenue for time series analysis might be to simply compare the power spectra of weather data with the climate models, if you can somehow extract that information from the models. I seem to remember that there was a Physical Review Letters article that did that and found that there was no similarity.

    • A chaotic systems has to be sampled at three times the rate of the longest period in the signal rather than twice the period if a Fourier series is what you want.

      Yes and there’s many indications from various papers that there are hundred and thousand year or more periods in climate, and we have decent data for most of them for about 135 years… (really good data for 36 years…)
      So like Fourier Analysis do you need to window the data as well? I note that Willis detrended the data but that’s not the same thing as windowing.
      I (and probably Willis) would appreciate a link to the common mistakes of Lyapunov exponent calculations. I know the ones for Fourier analysis pretty well. I assume there’d be a similar checklist.
      Peter

    • A more fruitful avenue for time series analysis might be to simply compare the power spectra of weather data with the climate models, if you can somehow extract that information from the models.

      I seem to remember that there was a Physical Review Letters article that did that and found that there was no similarity.

      Here’s a great paper at extracting any useful spectra and temporal spectra from SST. It shows there’s no useful information save ENSO. I’d love to see this same method used for all the other common climate data. Or maybe it’s time for me to learn R and see if there’s a decent wavelet library:
      http://paos.colorado.edu/research/wavelets/bams_79_01_0061.pdf
      IMHO I believe this paper invalidates all attempts to correlate temperature to geomagnetic, solar, (pick your favorite data), because correlating with noise is a waste of time…
      Peter

    • paullinsay,
      “A more fruitful avenue for time series analysis might be to simply compare the power spectra of weather data with the climate models, if you can somehow extract that information from the models. I seem to remember that there was a Physical Review Letters article that did that and found that there was no similarity.”
      I’ve read a number of papers in the literature on this. At short times scale (up to about a month) both the data and the models show red noise (power goes as 1/f^2) as expected for a chaotic process. At longer times scales (up to a few decades) both show near white noise (power independent of frequency) as expected for a random process. On even longer times (centuries on up) models continue to show near white noise while data show red noise. If you want to search for this stuff, a good name to start with is S. Lovejoy.

      • On even longer times (centuries on up) models continue to show near white noise while data show red noise. If you want to search for this stuff, a good name to start with is S. Lovejoy.

        Interesting. I’m slowly working a project to try and set confidence intervals for a signal against the “is it noise” null hypothesis, for both frequency and time domain.
        If you have something better than a google phrase I’d sure appreciate it. Modeling the noise spectrum of signal is pretty important, and if AR1 isn’t going to do it then I’ll need to work out another method.
        Also this says something subtly important about the climate models..
        Peter

    • Excellent response.
      “power spectra of weather data” It appears random. No one AFAIK has been able to show a low dimensional attractor for weather/climate data. Judith Curry and Marcia Wyatt did a great job with the stadium wave but that’s about it.

    • “A simple example is provided by experiments done on Rayleigh-Benard convection where a fluid cell is heated from below and held at constant temperature at the top. When the heating is small there is no convection. Once the bottom plate is warm enough simple periodic convection cells develop.”
      There is a small chapter on this in “Convective Heat Transfer” by Adrian Bejan, 2005. Some cell patterns are shown and formulae given for transfer rates etc. The gap between plates has to be pretty small compared with the area. Whether the atmosphere qualifies as analogous or not, look and form an opinion. Bejan feels the problem of heat flux through the atmosphere to space was so simple it was not even interesting and dropped the subject. The atmosphere has far more capacity to self-govern the temperature (storm thermals and clouds) than our capacity to perturb it. Plus ‘space’ is not a plate constraining thermals’ vertical height anymore than GHG’s are a sheet of glass.

  15. A beautiful question.
    Is the time span shown adequate to address that question?
    Given the planet cycles from ice age to not ice age,maybe it is bipolar instead.
    trying to remember Chiefio’s post on lunar cycles and long term planetary weather cycles.

  16. If you want to know what chaos is, ask Maxwell Smart. But in reality I can say just one thing regarding these findings. Mathematics can imitate reality but it can not reproduce it. It’s an analog world, not a digital one, so it comes as no surprise that real world chaos doesn’t necessarily reproduce in math. If the climate wasn’t chaotic, the models would have it projected already, we could read an almanac from some past year and know exactly what to expect 10 years from now.

  17. “Another group of datasets, those at the bottom in gold colors, are quite different from the random and the observational datasets. They are a variety of chaotic datasets. Note that they all share a common factor—as mentioned above, over time the rate of separation (as measured by the “stretching factor”) increases. The rate of separation doesn’t just go to a certain high point and stay there like the random or climate-related datasets did. The rate of separation in chaotic datasets continues to rise over time.”
    What does ‘time’ represent for those datasets? If you look at those at the small enough temporal scale, on some portions, it might look as not increasing.

      • I wonder of randomness is a better word than chaotic?

        Don’t confuse randomness with chaos. In English they sound like synonyms but in math they are entirely separate concepts.
        As noted in above posts, chaos is notable when there’s only a few variables, when the system is defined by approximately > 10 variables the system tends to be dominated by randomness. Which is yet another reason why I believe all “X is correlated with Y” for natural phenomena should include a Monte Carlo simulation against the correct spectrum of noise…
        (BTW Willis, you do get a similar curve with pink noise… right?)
        Peter

      • I was a student of the late George Marsaglia, an expert in pseudo random number generators and the tests they must pass, and he philosophically maintained that it was not possible to mathematically define our “intuitive idea of true randomness” as opposed to pseudo randomness for a sequence of bits uniformly distributed. I even argued with him about obtaining “true” randomness from natural sources such as a noisy diode or quantum-level phenomena, but he always maintained that any process for obtaining such bits would introduce biases.
        There is a website, Random.org, which provides “true” random numbers based on atmospheric noise picked up by radios tuned between broadcasting stations; but the website still italicizes the adjective “true” modifying “randomness.”
        So I guess my question is, until there is an agreed-upon mathematical definition for true randomness, (or a proof that such a definition is impossible, and hence its existence is axiomatic), how can we precisely define the difference that definitely exists between chaotic phenomena and random phenomena? –and what influence does one have on the other?

  18. I question the usefulness of your analysis. The Lyapunov exponent is defined in terms of the closeness of trajectories of the system. To define this you need to describe what the system is and what it means for two states of the system to be close to one another.
    In this context I would say that the climate as a whole is the system. Individual measurements of climatic variables are not the system. They just measure one dimension of it. Consider for example temperature at a single location. Two different days may have the same temperature at that location. But those days could be very different in other respects. One could be wet and stormy, the other fine and clear. In one the winds could be blowing from the east. In the other from the west. The future evolution of the temperature at that site seems to me to be likely to be much more strongly influenced by these variables you have not measured than by the inherent ‘stretchiness’ of temperature.
    In conclusion I don’t think Lyapunov analysis can be meaninfully applied unless you do it in multiple dimensions with a measure of closeness that uses all important variables and not only one of them.

    • “Individual measurements of climatic variables are not the system”
      That is correct but the dimensionality appears in all state variables and their derivatives. One can’t describe a system in term of one state variable and its derivatives but one can define the dimensionality of the attractor and hence decide if the system is chaotic or not.

  19. I threw a large rock in the ocean. How many times will I throw that rock? Will it be the same exact shaped rock? Will I throw it exactly with the same force and in the exact same position and in periodic intervals? Is that chaotic or random? It depends on what happens when convergence and divergence are happening. Next I’m going to blow bubbles in the ocean. Maybe I’ll splash around. Maybe the rock is only 0.04 meters wide, maybe it’s 4 km wide.

  20. Interesting post from Andrew Bolt this morning about PM Turnbull presenting an award to a sceptical sounding scientist. I hope this may fit this Willis post.
    Here is the post————–
    Warmist Turnbull gives prize to scientist for questioning warming catastrophism
    Icon – Comments 0 Comments | Permalink
    Andrew Bolt Blog Icon Arrow
    Andrew Bolt
    –, Friday, October, 23, 2015, (6:33am)
    Malcolm Turnbull, a warmist, has seen his Prime Minister’s Prize for Science go to Professor Graham Farquhar for his work on photosynthesis.
    Farquhar has won for work that actually challenges the apocalyptic global warming models we’re told not to question:
    Professor Farquhar has also worked to unravel the mystery of why wind speeds and evaporation rates appear to be slowing under climate change, a phenomenon that goes against most climate modelling.
    “We studied the meteorological records at all these sites and discovered from the physics of it that it was actually the wind speed going down that was causing the majority of decrease in evaporative demand,” he said.
    He said the discovery meant climate change could be a lot wetter than many people realised.
    That’s actually good.
    In fact, Farquhar sees even more good in man-made warming:
    “My reckoning is that if we could get rid of all the anthropogenic carbon dioxide emitted since the industrial revolution, then agricultural productivity would drop by 15%,” he says.
    (Thanks to reader Rafe Champion.)

  21. My guess (as a total amateur) is that the system is multi-cyclical, with temporal variances dependent upon the galactic and heliospheric orientation of the planet, plus chaotic events thrown in for the fun of it.

      • Let me restate that (as a plebe) and say that the ‘system of global climate’ appears to me to be multi-cyclical, with the frequency and amplitude of the various cycles governed by the orbital and heliospheric influences exerted upon the planet. probably still greatly oversimplified, but not so naively monocausal as the CO2 reradiation theory.

    • This also begs the question (for the greens) as to whether human proliferation is a “chaotic event”.

  22. The use by Warmistas of the term “chaos” is a political term, not a mathematical term. The political term chaos means “running around in circles waving your arms in the air and screaming we’re all going to die” which unfortunately is true. Willis here has done a nice expose of the mathematical term.
    It also suggests that sunspots and optical depth do not produce chaos in the system.

  23. There may be some evidence that the climate is chaotic but not on a time or length scale that you are using. Chaotic systems are predictable on time scales short compared to the largest Lyapunov exponent and because they are sort-of-kind-of periodic if they are low dimensional enough. This can be used to make predictions by analogy. For example, if you take the last two sunspot cycles and slide them back over the historical record, they line up fairly well with the sunspots just before the Dalton minimum around 1800. That seems to be not a bad prediction for what’s happening now with the sun.
    Joe Bastardi of Weather Belle has hinted that they do the same thing for their weather forecasts by looking back in the historical record for matches with the current state of the Jet Stream, sea surface temperatures, and so on. By using the best match they can get a prediction for the near term. Lorenz tried to do something like this in 1963 but certainly didn’t have either the data or the compute power to do a good job.

    • Thanks, Paul. If they are only chaotic on a long timescale, then why does the weather diverge from models within a few days?
      w.

      • That is a great question. An opinionated quasi answer. Because the weather models are unquestionably of the nonlinear dynamic sort. Per comments above, they contain time dependent feedbacks so are by definition mathematically chaotic. So, display sensitive dependence on initial conditions. Which initial conditions can never be sufficiently known, as Lorenz first showed.
        Whether analog real weather operates just like its math models is debatable. I am reminded of an Einstein dictum: ‘God integrates naturally. I have great difficulties’. Hope got that dictum about right since did not Google. Regards for a stimulating post.

      • Weather is only chaotic on some very small surface areas of the planet giving extremes, but overall the planet’s huge surface area is a very stable system. This is why weather diverges from models within a few days. That’s just one of many reasons why only using 0.1% of the planets surface is an awful way of measuring climate.

      • The time scale that matters is that of the largest Lyapunov exponent, actually its inverse. If it’s three days, after five days the prediction and the weather aren’t going to match well.
        I should have said back in time not time scale, my bad. I meant having data that covers long time spans so that you would have a chance for a pattern match. The “periodic” pattern you’re looking for is probably only a few days to a few weeks, a season? That is something that the meteorologists have a handle on. I don’t mean to bs you, I’m trying to just give an idea of what’s involved.

      • It’s not the time scale, it’s chaotic period. If you measured your weather to the milli-degree of temp, the 1/100 of a percent of humidity and the microbar of pressure, you would notice the actual weather diverges from the model after a few seconds, rather than a few days!

    • “If they are only chaotic on a long timescale, then why does the weather diverge from models within a few days?”
      Too many damn butterflies ; )

      • “If they are only chaotic on a long timescale, then why does the weather diverge from models within a few days?”
        Too many damn butterflies ; )

        Ha, probably true metaphorically speaking.
        Perhaps the weather/climate system is chaos short term, randomness mid-term, and back to chaos long term. Can systems be chaotic xor random on different time scales (window sizes)?. I think the answer is yes, but the math is beyond me, so it’s a hunch.
        Peter

  24. Re: Is the Climate Chaotic? 10/22/15:
    This subject was a topic on WUWT on March 15. That discussion covered the questions of chaos and linearity in the climate based on external, established definitions of the two terms. But Willis immediately takes the discussion into the weeds of Lyapunov exponents and stretching. I say into the weeds because the dissertation has zero references or sources. Of course, an author can always insert his own definition, and Willis’ definition is this:
    A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.
    But what are the “states” of the climate? Next, what are the “initial states”? Can we just pick whatever time we want for the “initial states”? Might it be an ice age or a so-called genial epoch? But why the plural “states”? What is the other one? What does “nearby” mean? Is that a distance in space, or is it in time? How do measure distance to know if these states converge or diverge?
    Chapter 6 on the chaosbook.org website is about Lyapunov exponents. Part I of that book puts chaos in the context of basic notions of dynamics, prescribing 7 initial steps. The first step is, “Define your dynamical system (M,f); the space of its possible states M, and the law f^t of their evolution in time.” How do we do even get started for climate in order to apply Lyapunov anything?
    Then in Chapter 6, we have, “This sensitivity to initial conditions can be quantified as [see hard to type equation in html] where lambda, the mean rate of separation of trajectories of the system, is called the leading Lyapunov exponent.
    The path through the weeds produces exactly the same problem as discussed back in March. Lyapunov exponents and stretching, like the concepts of linearity and chaos, apply only to the equations of a system, not to the real world the scientist hoped to represent by his choice of equations.
    The climate is neither linear nor nonlinear. It is neither chaotic nor well-behaved, and the relevant questions, like the conclusions of IPCC, are competent only with respect to the models, the system of equations.

    • Some people can’t tell the weeds from the forest. It’s not his fault, in a way. It’s our fault for not calling shenanigans.

  25. By the way, just because of a low dimension analysis of the system, it appears to be random, not chaotic, it does not mean it is random and not chaotic.
    You have to add dimensions from the state space and look again. And again. Until you find chaos or you find your dataset is too tiny to identify close to identical states. The later is actually the case. The case where you exhaust all dimensions I won’t take into account for obvious reasons.
    There is more: Pseudo-temperature does not define the physical state of the system. Pseudo-temperature is not even part of the state of the system. To look at it to identify if the system is chaotic or not is very wrong.
    Now, the big claim:
    Not only the dataset is too small, but even if the dataset for the whole climate since its inception would be available, it would be still too small. The claim is that the system never ever came with its states close to an old trajectory in the state space, so even with the over four billion years of the whole climate data available, you cannot identify two close states to start comparison. The system is so complex that even with the huge time passed it had time to explore only a tiny portion of its state space and it never ever came back close to an old state. It might appear so if you use a tiny subset of the dimensions, but probably you’ll get the ‘random’ conclusion.
    Comparing an insanely complex system with some very, amazingly simple mathematical systems is not ok. Not at all.

      • Somebody gets it. And somebody ain’t Willis.
        Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
        So simple. So … chaotic.

    • He doesn’t understand and finds “these results were a great surprise to me.” Problem is, he still doesn’t understand.
      I bet he will confuse auto-correlation again.

    • “You have to add dimensions from the state space and look again. And again.”
      Adding dimensions to the state space and looking again was the first practical application of chaos theory I ever encountered. Add some pattern recognition to the procedure and it can be automated. It seems to be the mathematically terminal result of all the chaos blah, blah, blah.
      The problem is, people are looking for some great theory, some great truth from chaos and it just isn’t there. I looked for it and ended up being very disappointed. I’m like, “That’s it? Plot the dimensions that contain the information and then look for patterns? That’s it? Ugh.”

    • Pseudo-temperature is not the system. Climate system, if you want to talk about a sub-system of the whole, with some numerology applied. But even so, with a large enough temporal scale, you might have some (some would say tiny, using that temperature scale) chaotic surprises. Nevertheless, pseudo-temperature is not the climate.

      • Temperature is not the best way to measure energy, but it is the best we currently have. At least with Kelvin at absolute value, energy and temperature has a better meaning. This gives a better indication how changes we have seen over that past century+ are greatly insignificant compared to overall energy in the climate system, where zero Kelvin represents none.

  26. I hope this sort of fits here as well. The 2014 Concordia Uni study found that there has been about 0.56 C of warming since 1800 attributable to the top 20 countries. Therefore according to this study total warming would be about 0.68 C over the last 215 years.
    Not a lot of warming considering the planet’s recovery from one of the coldest periods over the last 10,000 years. I’ve read that many studies show that glacier advance was at the highest Holocene point during the LIA.
    So how much of this slight warming is attributable to human Co2 emissions and why? I ‘ve read that the IPCC states that attribution starts after 1950. Here’s a reference to the study————–
    http://berc.berkeley.edu/ranking-global-warming-contributions-by-country/

  27. I like the analogy of a straightjacketed crazy man in a rubber room. A lot of the time he sits around and drools. But then some tiny event in his brain stirs him into a frenzy…he rises, screaming, and runs about, bouncing off the walls, till he is exhausted. Then he settles again into a good long drool.
    The Earth’s climate, i.e. weather averaged over time, is dominated by negative feedbacks. This is a necessary conclusion based on the relative stability of the climate over eons. The negative feedbacks are like the walls of the rubber room. Perhaps a chaos theorist would call them strange attractors, as in comments above.
    Within the room, however, the crazy man is a chaotic system. Tiny instabilities in his brain, perhaps even a single synapse or two firing at random, can produce diverging (stretched) behavior of comparatively large magnitude. One moment he is sitting, the next bouncing off the west or the north wall. Yet the walls keep him in within a certain locus.
    Thus the weather may be chaotic, but the climate overall, bounded as it is by the negative feedbacks of clouds, storms, ocean heat sinks, etc., is far less so.

  28. Sometimes I wonder if Man’s ego calls something “chaotic” as an excuse. If Man doesn’t really know what’s going on, call it “Chaotic”. If Man can put a name to what is beyond him then, somehow, he really does know. After all, Man gave it a name so Man’s “on top of it”.

    • GD, mathematical chaos is precisely defined as a consequence of non-linear dynamic systems (having time dependent feedbacks, for the uninitiated). Now, whether and how that abstract math applies to Earth is a discussion worthy subject.

      • Thank you.
        So in Math “chaos” has a literal meaning. Beyond that it’s more of a figure of speech or a … comfort? … as I used it.
        Sort of like “imaginary numbers”. Are we talking Math or climate science? 😎

    • I think you are onto something here. Chaos has certain characteristics that identify it and I think Lorenz’s discovery of it is viable, but I suspect that the term gets broadly used when we can’t quite figure something out satisfactorily. I notice in discussions on climate, the word it self is repeated and hurled about and coupled atmosphere/ocean/land complexities are bemoaned – it is a kind of sophisticated hand waving thing. I’ve questioned this before – usually following a dissertation by Dr. Robert Brown on the intractability of determining where climate is going.
      Having the luxury of not being taxed with resolving such issues, I just offer that for such chaos, somehow there seem to be only a few kinds of weather (I reckoned 10 in a comment above where I live, probably half that in the tropics) and each time I’ve seen them, they look pretty much like the last 1000 times I’ve seen it. A sunny day at the beach on a clear day in 1950 looks like a carbon copy of the same thing in 2015. A rainy April in Ottawa, Ontario is indistinguishable from one year to another. Heck, we had a thread on here many months ago that talked about the few days earlier the cherry blossoms were blooming in Japan in 2014 compared to 50 years ago.
      Look at the polar ice graphs they pretty well waggle up and down a bit more or a bit less and change direction on September 22nd at both poles!! Farmers get caught once in a while planting a week or two early but generally agricultural output follows pretty smooth curves. The first frost, the first and last snows, the tropical storm and tornado seasons….a nor’easter bring rain, nor’wester cold and snow in the fall to spring, pineapple expresses, Colorado lows, ………even migratory birds get it pretty well right and monarch butterflies begin to collect in the Niagara Peninsula in Ontario in the fall where they wait for the never failing north wind to assist them in their flight over lakes Ontario and Erie. How chaotic is that?

      • Is The Climate Chaotic?
        Word has it;
        ” While the earth remaineth, seedtime and harvest, and cold and heat, and summer and winter, and day and night shall not cease. ”
        ~ Modeler of modelers ; )

    • Gunga has just defined a large percentage of what we currently call science.
      But the concept of chaos in not that complex. Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
      In other words, the fine grain of our future state is unpredictable and unknowable. Anyone who can’t deal with that has yet to grow up. Chaos is what makes life interesting. If the future were predictable we would all be very bored.

      • Unless we had a copy of “Grey’s Sports Almanac”. 😎
        (Reference to “Back to the Future” for those I just confused.)

  29. As long as there is some kind of tax on the air being considered, then I feel better.
    I’m sure the largest emitters will pledge …something.
    It’s all good, didn’t even have to fly all those private jets to the coven.
    Nobody notices expenditures like that.
    Nobody.

  30. Willis
    Far too many people equate temperature with climate. Temperature is merely one of the variables that make up climate (others include humidity, precipitation (amount, nature and cycles thereof), air pressure, wind, windiness, influenced by El Nino/La Nina conditions etc etc). Each of these parameters is variable, and constantly varying within bounds, and each may interact with each other.
    I am far from convinced that you have looked at climate. It seems to me that you have only looked at a couple of the parameters (predominantly temperature) that go into the climate melting pot, and thus you are not in a position to say that climate is not chaotic.
    That said, your findings are interesting.

  31. The path of least resistance allows a degree of randomness. The issue is, how wide is the path. That is likely bounded by statistics related to regional extremes combined with broader regime shifts.

  32. “If you drop two sealed empty bottles overboard in mid-ocean, one on each side of a boat, they will drift apart over time. This separation will be slow at first, and then faster and faster as the two bottles encounter different winds and currents in different areas.”
    Man, I don’t like that definition of chaos. So the bottle’s separation is at first not chaotic and when it reaches the magic exponential threshold it suddenly becomes chaotic? A black hole is chaotic?
    No. This is that Lorenzian statistical chaos that drives me nuts. Granted, it has some impressive practical applications in information theory. Our cell phones work because the medium is the message.
    Every message must have a medium. The waves in the ocean would not exist without the water. But just because impressive practical applications can be derived from a particular definition of medium and regressed in that framework does not mean that the medium or the regression is fundamental.
    Take Isaac Newton. My God, what a medium he construed, and what incredible derivations followed from that. Yet we now believe his medium is fundamentally untrue.

  33. I recommend reading the climate research being performed by Dr. David Evans at the following location. At the present time there are 14 articles which will be followed by more. These articles contain semi_complicated math but the theory is presented in bite size chunks. The data presented by Dr. Evans accepts all the IPCC values and then shows how the models are incorrectly constructed. He states that there is an intellectual mind block because warmest state that the math is dead on while the sceptics state that observations invalidate the math.
    http://joannenova.com.au/tag/climate-research-2015/

  34. It turns out that even textbooks devoted to chaos do not really define the term. Wolfram

    Many years ago I heard an interview with Edward Lorenz (the father of chaos theory). He described his aha moment.
    Lorenz was running a climate model and needed the results for a conference. Something went wrong before the model could finish its run and the results were incomplete. He decided that if he re-ran the model with fewer significant digits, it would run faster (and he would have something to present at the conference). His results wouldn’t be as accurate but they would be a sufficient approximation. What he actually found, however, was that the results were completely different.
    As far as I can tell, Lorenz defined the climate as the very prototype of a chaotic system.
    1 – The climate is exquisitely dependent on initial conditions.
    2 – It is impossible to define the initial conditions with sufficient accuracy.
    3 – It is impossible to have enough significant digits to compute an accurate result.
    4 – Therefore, changing anything (even slightly) about the inputs to the model results in completely different outputs. butterfly effect

  35. Nature is not chaotic. You can make a intelligent guess if the apple will fall down or up or if it will snow or rain. Natural laws give some limits of variations and even within those limitations You can predict fairly good by observing and calculating.

    • I don’t think that’s true. Look, for instance at water droplets from a dripping tap (a description of how they are chaotic appears above). Or snowflakes, shorelines, thousands of possible examples ranging right down to the fingerprints on your hand.

    • I think you may have your own definition of chaos. With the other definition of chaos, the scientific one, most natural dynamical systems (non linear, not at equilibrium) are chaotic. You may find many exceptions with simple ones, but there are plenty of simple ones that are chaotic, too.
      To be chaotic means: to be sensible to initial conditions, to have dense periodic orbits and to be topologically mixing. It does not matter that the state space is bounded. It does not matter how you perceive the distance between bounds. With a change of scale it might appear tiny, anyway. Look what Matt did with the pseudo-temperature not so far above. Despite that, it does not show that the system is not chaotic.
      Sometimes, you may get away with a change of scale. For example, even a gas with a well defined and unchanging temperature (that is, at equilibrium) is indeed chaotic. Just look at what molecules do.
      But the macro system appears not to be. Of course, in this case there is the luxury of being at equilibrium, a luxury that real natural systems seldom offer.
      Put that gas out of equilibrium, make it flow and have temperature and pressure fields rather than the same everywhere and things get quite interesting indeed. And quite hard to predict for a long time, except some very particular and simple setups.

  36. I am not sure, but I get an uneasy feeling that you have not got the right definition of chaotic – more like a definition of ‘unbounded’.
    Divergence is not the same as chaos
    Nor it seems is Lyapunov a criteria of chaos, but of stability.
    https://en.wikipedia.org/wiki/Lyapunov_function
    And the exponent that shows divergence is indeed an indicator of chaotic behaviour, but the converse is not true. Non divergence does not indicate non chaos, as many chaotic systems are bounded and represent chaotic orbits around attractors.
    I fear all you have done here is prove mathematically that Climate is (reasdaonably) stable and governed by some kind of overall negative feedback, which is pretty obvious given the fact life exists at all.
    I suspect its quite difficult to come up with a metamathematical test for chaos – especially chaos with many variables.
    One approach is of course Fourier analysis. But quasi periodic chaotic behaviour might show up there too as a peak in the spectral density.
    I deficiently applaud the intention shown in this article, but I fear – though I do not have quite enough maths to prove – that you have missed the target a bit.
    Chaos mathematics has not been developed very much and its not a popular subject, because it doesn’t give answers. by and large.
    But this attempt shows we do need a test for regions of chaos so we can mark them with ‘Here Be Dragons: Give up or die” and move onto more profitable endeavours.
    I just dont think Lyupanov is the acid test.
    http://www.hindawi.com/journals/mpe/2010/720190/
    us a very interesting paper.
    Lyapunov exponent is the most important quantity to chaotic systems as a positive maximal Lyapunov exponent is a strong signature of chaos. In the contrary, a zero maximal Lyapunov exponent denotes a limit cycle or a quasiperiodic orbit and a negative maximal Lyapunov exponent represents a fixed point.”
    Which I think says what I was trying to say. if its positive its chaotic but if its zero its not necessarily non-chaotic.
    Anyway that paper – which I found before, asking a similar question, had one important effect on me personally. I realised I dont have the maths or the time to acquire it to answer the question.
    My humble point is to suggest that you too may not have quite enough maths, either, which shouldn’t be taken as a criticism, because there seem to be only half a dozen people in the world who do.
    It actually us a compleat bitch of a problem. And requires a mathematical genius or three to answer.

    • My humble point is to suggest that you too may not have quite enough maths, either, which shouldn’t be taken as a criticism, because there seem to be only half a dozen people in the world who do.

      So many papers are written, peer reviewed, and published whose authors suffer from exactly the same problem. Folks with a weak understanding of statistics or signal processing will throw data into a random tool and think they have meaningful results. It’s pathetic but the really pathetic part is that it gets past the editors and peer reviewers. Here’s my favorite example …

      … the study was 100 percent authentic. … We ran an actual clinical trial, … the statistically significant benefits … that we reported are based on the actual data. … The results are meaningless, … Chocolate helps weight loss

      If one looks at enough factors, any data set will produce correlations. Naturally, the authors avoided the participation of any genuine statisticians; that would have spoiled the hoax. It’s fairly obvious that any paper that uses any mathematical analysis should involve real mathematicians. That should have stopped Michael Mann dead in his tracks.
      Engineers understand that they should practice within the limits of their knowledge, skills and abilities. Many scientists seem to lack that understanding. [/rant]

  37. The opposite of what I had assumed. There is hope therefore that Climate and Weather Science may yet achieve to level of understanding and predictive capability.

  38. I am puzzled by your methodology & conclusion.
    In order of predictability the ladder is
    constant > periodic + linear > “simple” non linear > chaotic > random.
    So, regarding methodology, to answer the question “Is The Climate Chaotic?”, I would
    * since “weather is not climate”, I wouldn’t use weather data, but climate data : yearly averages of rain, solar irradiance, number of days with rain, freezing temp, high speed winds… Climate is much more predictable than weather, as the saying goes “climate is what you expect, weather is what you get”.
    * indeed use, as comparisons, a single random (one is enough) and a couple of known chaotic maps, but also some more predictable maps (sunrise hour at your home for instance). I would also add some funnies : my heartbeat data, Dow Jones, population of New York …
    And, regarding conclusion, after your work I wouldn’t say that weather/climate variables are “weakly chaotic”, but, quite the reverse : so “highly chaotic” that they compare more to random than to usual chaotic reference such like logistic map. But then again that’s because you used weather variable, not climate. I guess climate will to much less random.

  39. Natural laws tend to give limitations of variations and even within those limitations. It is easy to predict by carefully observing and calculating.

  40. Beyond the difficulty in merely agreeing upon a definition of “chaos” I believe you are making a critical mistake by trying to find chaos in data sets of single variables (like daily temperature at a single location or lowest annual Nile-o-Meter reading). I would expect a chaotic system to produce data sets exactly as you found–highly random particularly when time frame of the data sets is either very short relative to what appear to be profound chaotic events in the system (like ice ages) or very long relative to the sublime chaos over a very short period of time (say a single day or night of temperature at a specific location sampled by the millisecond).
    The inclusion of the Nile-o-Meter in your piece is a curious one. While I have no idea if there is anything like a continuing record, there is still at least one functioning ancient Nile-O-Meter in Egypt at the temple of Dendera. (I saw it this year.) Yes, I am fully aware that the construction and operation of dams on the Nile have profoundly affected the natural fluctuation. Even more profound effects are likely to occur in the near future when the construction of another dam on the Nile (this time in Sudan) is complete. If a reasonably continuous record is available that begins in ancient times and continues to the present day, I suspect that your analysis of the data set would be quite different.

  41. Summary of definitions of chaos above:
    ” a system wherein nearby initial states either converge or diverge exponentially.
    It is time dependent feedbacks that define chaotic systems.
    Systems are chaotic if very small difference in initial conditions yield widely different outcomes.
    If the climate wasn’t chaotic, the models would have it projected already,
    to be sensible to initial conditions, to have dense periodic orbits and to be topologically mixing.
    If Man doesn’t really know what’s going on, call it “Chaotic””
    Me, I’m partial to the last one. Possibly the only verifiable initial condition is the big bang?

  42. I’m surprised that any thinking person would assume that the climate is chaotic when it is trivially obvious that it is not- at least for the last 6 -700 million years. During this time the temperature and the oceanic and atmospheric chemistry has stayed within the narrow limits required for carbon based organic life.
    We can also e.g. back cast the ephemerides for about 60 million years before the differences in how different computers round off decimal points causes divergence.
    There is evidence in the geological record for the 405,000 year eccentricity cycle more than 400 million years ago.
    The millennial solar activity cycle which has probably just peaked in 1971 has been identified in the Holocene and as far back as the Miocene.
    http://3.bp.blogspot.com/-NuOJUXIC050/U9A9WuN2thI/AAAAAAAAAUg/o2nWMaYMBlY/s1600/KernMioHolo.png
    This does not mean, however, that climate can be forecast using reductionist climate models.The modelling approach is inherently of no value for predicting future temperature with any calculable certainty because of the difficulty of specifying the initial conditions of a sufficiently fine grained spatio-temporal grid of a large number of variables with sufficient precision prior to multiple iterations. For a complete discussion of this see
    Essex: https://www.youtube.com/watch?v=hvhipLNeda4
    Earth’s climate is the result of resonances and beats between various quasi-cyclic processes of varying wavelengths combined with endogenous secular earth processes such as, for example the changing geomagnetic field.. It is not possible to forecast the future unless we have a good understanding of where we are at present time in relation to the current phases of these different interacting natural quasi-periodicities which fall into two main categories.
    a) The orbital long wave Milankovitch eccentricity,obliquity and precessional cycles which are modulated by
    b) Solar “activity” cycles with possibly multi-millennial, millennial, centennial and decadal time scales.
    The convolution of the a and b drivers is mediated through the great oceanic current and atmospheric pressure systems to produce the earth’s climate and weather.
    After establishing where we are relative to the long wave periodicities , especially the solar millennial cycle, to help forecast decadal and annual changes, we can then look at where earth is in time relative to the periodicities of the PDO, AMO and NAO indices and based on past patterns make reasonable forecasts for future decadal periods.
    For the latest forecasts and complete discussion see
    http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html

  43. The reference for the Figure is
    Fig.6 Kern et al http://www.sciencedirect.com/science/article/pii/S003101821200096X

    What are the units of the vertical axis in the figures above and what measurements were they derived from?
    I read the paper, and as far as I can tell, you won’t be able to tell us the units because there are none. The data is derived from a total of 3 proxies (e.g. bore drill of clay in Austria), and then munged through numerous processes such as arcsin transform. By the time you are done, it’s unrecognizable and the peaks could mean anything as they might just be aliasing or the result of some nonlinear transformation on random noise. I was unable to figure out the final units after all that munging. Please clarify, I might have missed something in the paper.
    I do appreciate the use of confidence level bars, but the paper doesn’t say how they were created. Monte Carlo against appropriate spectra of noise is my assumption from your references, but it’s unclear.
    I like the analysis approach of looking at spectra and wavelets, but the proxy munging makes me cringe a bit…
    Peter

    • what are the units of the vertical axis in the figures above and what measurements were they derived from?

      Sorry, I read too fast. I understand the vertical units now (gamma ray CPS from your radiameter instrument). I believe the standard is to also measure for the null hypothesis by running one that is shielded from the experiment but otherwise exposed to the outside world. I didn’t see where that was done. Something that tripped up the cold fusion folks I believe…
      Still think you need more proxies and a reproduction…
      Peter

  44. I think we can all agree that the Earth’s climate is a complex system with many poorly understood positive and negative feedback loops. However, one can deduce a very important characteristic from its behavior. Dynamic systems are either stable or unstable. Dynamically stable systems have negative feedback loops that return the system to equilibrium (picture a ball inside a globe, it can oscillate but it returns to the bottom). In an unstable system, a change in one variable can cause the system to obtain a new equilibrium point (picture a ball on top of a globe, a small shove and it falls off and obtains a new equilibrium). The theory of man-made global warming relies on the assumption that the earth’s climate is an unstable system. Global warming advocates argue that a 3% change in one minor system parameter (CO2 concentrations) causes the system to achieve a new equilibrium. However, the earth has experienced changes in solar radiation, volcanic eruptions, great meteor strikes, and even significant changes in CO2 concentrations, yet the earth’s average temperature has fluctuated just 18 degrees F over the last 600 hundred million years. The earth’s climate is clearly a stable system and if it were not, the earth would have fried or frozen eons ago and climate change advocates along with the rest of the human race would never exist.

  45. Thanks, Willis. Very interesting essay.
    You write:
    “Now, these results were a great surprise to me. I’ve long believed, without ever checking it, that the climate was chaotic … however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest.”
    W.M. Briggs and Roy Spencer seem to believe otherwise and I agree with them. But I’m looking forward to your developing understanding of this matter.

    • Willis’s analysis does NOT show “not chaotic in the slightest”. Willis’s analysis shows that he doesn’t know the difference between random and chaotic. He then conflates the two, sees only randomness then concludes that there is no chaos.
      This is in direct contradiction to Judith Curry’s work that DOES show a stadium wave effect affecting climate. Judith knows what she’s talking about. Willis does not and he misleads people.

  46. Willis,
    The standard definition of climate implicitly assumes that climate is not chaotic. Otherwise you could not define climate properties in terms of means and standard deviations. To have any hope of creating a climate model requires that climate not be chaotic, otherwise you could not have any hope of modelling it.
    Weather is chaotic. But if a chaotic system is bounded, then the Lyapunov plot must eventually approach a constant value, since the distance between trajectories can not increase forever and stay within the bounds. So if you take a Lyapunov plot for such a system and plot it on a suitable timescale (one that is long compared to the time needed to approach the constant value), it will look like a plot for a random number (rapid rise to a constant). For temperature, the rise takes place over weeks, so it is not surprising that a Lyapunov plot for monthly and globally averaged data looks random. During the initial rise, the initial value is important, for the constant part of the plot, only the boundaries matter. That is what climate modellers mean when they say that weather is an initial value problem but climate is a boundary value problem. If right, that makes the statistical properties of climate predictable (at least in principle) even though the weather is not predictable beyond maybe a week. Lorentz called that predictability of the second kind.
    The problem, I think, is that the models focus on the atmosphere; but one of the boundaries for the atmosphere is the surface of the ocean. This following is speculation on my part.As long as that surface stays reasonably constant (or periodic), the climate should be constant. But the ocean has its own internal chaotic dynamics. Those processes occur on a much longer time scale (decades to millennia) than atmospheric chaos (weeks or less) so on sufficiently long time scales the climate is indeed chaotic. But your analysis does not pick that up since the instrumental record is not really long enough. Then when you get to millennia, the ice caps get in the picture and push the chaotic behavior out to even longer times scales.
    The modellers say that since they include the ocean in the models, the only boundary that matters is the top of the atmosphere. But that only works if they get the internal dynamics of the ocean right. I think that the failure of the models to get the sort of variability indicated by the Medieval Warm Period or the Little Ice Age (or even the pause) is an indication that the models underestimate the chaotic variability in the oceans.

    • The climate models on which the entire Catastrophic Global Warming delusion rests are built without regard to the natural 60 and more importantly 1000 year cycles so obvious in the temperature record. The modelers approach is simply a scientific disaster and lacks even average commonsense .It is exactly like taking the temperature trend from say Feb – July and projecting it ahead linearly for 20 years or so. They back tune their models for less than 100 years when the relevant time scale is millennial. This is scientific malfeasance on a grand scale.
      The temperature projections of the IPCC – UK Met office models and all the impact studies which derive from them have no solid foundation in empirical science being derived from inherently useless and specifically structurally flawed models. They provide no basis for the discussion of future climate trends and represent an enormous waste of time and money. As a foundation for Governmental climate and energy policy their forecasts are already seen to be grossly in error and are therefore worse than useless.
      The Fig below shows the sort of irrational thing the modelers do when they project a cyclic trend in a straight line..
      http://4.bp.blogspot.com/–pAcyHk9Mcg/VdzO4SEtHBI/AAAAAAAAAZw/EvF2J1bt5T0/s1600/straightlineproj.jpg
      For a better way see
      http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html

      • Dr. Page,
        I have never seen a convincing demonstration of 60 or 1000 year cycles. Even if there, such cycles account for no more than a small fraction of the variance in the temperature record, so they are no good for forecasting.

  47. I am skeptical of this definition of chaotic (and therefore the conclusions drawn). It fails to account for non-linearity, complexity, and feedback mechanisms, to name just a few factors. I need to learn more about this, but even textbooks on chaos theory and non-linearity avoid defining the term. See: http://mathworld.wolfram.com/Chaos.html

  48. Hi Willis,
    A couple of minor comments. First I very recently decided that I just didn’t know enough about dynamics (where nearly all nonlinear dynamical systems are at least potentially chaotic and where, as it has been said, studying nonlinear dynamics should be like a biologist studying non-elephant mammals — the exception rather than the rule). So I’ve bought three fairly serious books on the subject and am working my way through them. Because I like to watch, I’ve set up a directory in which I am building a sequence of projects in octave to exhibit e.g. logistic chaos (which is a bone-simple iterated map with a very clear chaotic regime, perfect Feigenbaum tree, etc) and other chaotic systems. Following notes and comments by Sprott (who has a superb website devoted to chaos and fractals that accompanies his equally excellent book Chaos and Time-Series Analysis) one of the first things I did was to take simple ODEs known to have NON-chaotic solutions and integrate them with a large fixed stepsize using Euler integration, verifying that for a sufficiently large stepsize they develop strange attractors and become chaotic (something I’d pointed out to Nick some time ago, but I wanted to see it for myself).
    By simple ODEs, I mean absolutely trivial ODEs that move a point around in a circle, a Van der Pol oscillator, things like that. The point being that even well-behaved systems of ODEs with perfect closed analytic solutions can become chaotic iterated maps if one integrates with too large a stepsize. As Sprott points out, this is likely to be true for many, many systems of differential equations, including linear ones.
    I spent today integrating the rigid driven pendulum, a second order nonlinear ODE with a well-known chaotic regime. Or, perhaps I should say, with a near infinity of not-so-well-known chaotic regimes. My purpose there was to investigate just how sensitive results were to stepsize using a good ODE solver with tight control of tolerance — Octave’s built in lsode with absolute and relative tolerances set to 10^{-9} and full use of adaptive stepsize (but with a user-selectable grid that can force a moderately finer granularity). Note that this is a much, much simpler system of only two coupled ODEs (to solve a simple nonlinear second order equation). Since even laptop computers are at this point supercomputers, even using octave instead of C to do the solutions (probably a factor of 10 loss of speed) let me play with millions of timesteps (plus adaptive solution between timesteps) across a transient and out to where the solution becomes either quasi-periodic or at the very least non-repetitive if not chaotic. Much to my interest and surprise I learned that this enormously simple system has multiple chaotic regimes when one varies a single parameter, e.g. the driving torque amplitude, interspersed with regimes of perfectly repetitive cyclic behavior (rarely pure oscillations but some N-fold cyclic oscillation that is apparently stable). The cyclic regimes display nothing like period doubling — one can run the system with a torque amplitude of 0.778 and get one behavior with a clear double cycle and run it for 0.779 and get chaos and run it for 0.780 and get a different periodic cycle. The solutions are equally sensitive to butterfly effect perturbation of initial conditions, the number of timesteps in the integration with fixed time intervals, pretty much everything.
    From this I conclude that it is not at all implausible to assert that the fond hope of the GCM modellers that using a stepsize 30 orders of magnitude larger than the Kolmogorov scale will even qualitatively reproduce the distribution of possible future climate trajectories may be unreasonable, as I think has been pointed out before by several people but without quantitative support. I’m hoping to set up a simple open system with some of the features of a “climate” and run it as an iterated map to see if the system develops a chaotic behavior for certain operating regimes. From what I’ve already learned from my reading, it seems very likely that it will.
    To get back to your observations above — I wouldn’t take them too seriously as evidence that the climate is or is not chaotic. This is where having some models (like the rigid oscillator) to play with can help a lot. If I just plot the phase angle of the oscillator against time, I can get a dazzling array of possible behaviors well out beyond the usual “transient” expected for the damping rate. In fact, I often run for times of 100,000 or 200,000 ‘units’ (on the scale of the natural frequency of the oscillator) because sometimes after 150,000 timesteps the system “discovers” an apparently stable quasi-period oscillation. Other times it finds one after 50,000, it persists for 30,000, and then turns back into chaotic oscillation. It isn’t even possible to do scaling of this stuff — change one digit in the sixth column in the inputs and you get completely different behavior.
    The chaotic parts can easily look just like the timeseries we have for temperature over a fairly renormalizable set of time scales. The problem is that those time scales may not be large enough to display any sort of divergent or phase space covering behavior. It is also difficult to map out a strange attractor with just one variable and time — the “chaos” is probably evident only on a torus in some (probably large) dimensionality so that when you look at global average temperature in particular, you are seeing only a projective view of the torus into a single dimension, nothing much to see and (for a bounded torus) no divergences per se.
    So you are looking at the wrong thing, in the wrong dimensionality. My recommendation is — invest in Sprott. It’s worth it. Learn more about dynamical chaotic models up front. I still don’t know enough to feel comfortable with chaos and statistics and dynamical maps, and I’ve spent many a fond hour integrating ODE systems that exhibit collectively nonlinear behavior and catastrophes in an earlier life. We both probably need to study — a lot — before asserting any sort of conclusion based on a simple glance at a short timeseries.
    rgb

    • Thanks, Robert, always good to hear from you. I learn more from your comments than just about any.
      I thought I was being cautious with my conclusions. What I can say is that I used a method that very clearly separated the datasets into three groups—one chaotic, one containing random data and also observations, and one containing a couple of forcing datasets (sunspots and volcanoes).
      Not only that, but that analysis agrees completely with a separate analysis I’ve done but not posted yet, using the Hurst exponent to differentiate between chaotic and non-chaotic datasets.
      So do I think that this is the final word? By no means. I just find it quite interesting, in that I had expected at least some of the six datasets I investigated to show some sign of chaoticity.
      How accurate is the method I used? As you say:

      It is also difficult to map out a strange attractor with just one variable and time — the “chaos” is probably evident only on a torus in some (probably large) dimensionality so that when you look at global average temperature in particular, you are seeing only a projective view of the torus into a single dimension, nothing much to see and (for a bounded torus) no divergences per se.

      Mmm … you say that I’m looking at “only a projective view of the torus into a single dimension”. Actually, that’s not true. The single dimension is “embedded” into a higher dimensional space. All of those were embedded into a 3-dimensional space. So I do need to look at using larger embedding dimensions. However, the simplest analysis has done well at identifying known chaotic datasets using just an embedding dimension of 3 … hang on …
      OK, I just looked at the results using embedding dimensions up to 8 … very little difference in the results.
      Also, I just looked at determining the optimum embedding dimension using the method of false nearest neighbors, and the answer is an embedding dimension of either 3 or 4 for all of the datasets, natural and chaotic. By that, I mean that after 3 or 4 dimensions, adding further dimensions only marginally improves the analysis. (Of course, for the random datasets the embedding dimension is meaningless.)
      So my question would be … since this method I’m using clearly and correctly identifies the various chaotic datasets, including Lorenz’s dataset which was based on atmospheric dynamics, and it clearly and correctly identifies the random datasets, then why does it identify the natural datasets as not being chaotic?
      It may be, as you say, that the datasets are too short. But chaotic is chaotic, whether it is short or not. I just re-ran the analysis using only N=663 data points for each of the chaotic datasets … and the results were basically unchanged.
      Again regarding time, that’s one reason why I used different datasets. The longest in time was a 1,200 year annual dataset (AOD) The smallest N was the annual Nilometer data, N=663. I also showed daily data, both average daily data for a small area (CET) and single-station data (Armagh). Both of these had N ≈ 50000. I had thought perhaps one or the other would be chaotic.
      Please don’t get me wrong. I’m feeling my way here. All I’m saying is that by the test I showed above, the six natural datasets are not looking chaotic. And I’m well aware that there are more types of chaos in Heaven and Earth, Horatio, than are dreamed of in my philosophy …
      In any case, you raise further interesting points here:

      To get back to your observations above — I wouldn’t take them too seriously as evidence that the climate is or is not chaotic. This is where having some models (like the rigid oscillator) to play with can help a lot. If I just plot the phase angle of the oscillator against time, I can get a dazzling array of possible behaviors well out beyond the usual “transient” expected for the damping rate. In fact, I often run for times of 100,000 or 200,000 ‘units’ (on the scale of the natural frequency of the oscillator) because sometimes after 150,000 timesteps the system “discovers” an apparently stable quasi-period oscillation. Other times it finds one after 50,000, it persists for 30,000, and then turns back into chaotic oscillation. It isn’t even possible to do scaling of this stuff — change one digit in the sixth column in the inputs and you get completely different behavior.

      Now, that makes perfect sense to me. I learned from Bejan and the constructal law that natural heat engines speed up until they run slightly into the turbulent regime. It seems like what you are talking about amounts to the same thing, that nature is running at the edge of chaos.
      What’s new for me in what you’re saying is the idea of the cyclical venturing into the turbulent, chaotic realm and then returning to laminar flow. I like that.
      Without conscious consideration, I’d always envisioned that a natural heat engine would run a bit into the turbulent realm, and then somehow just stay there, up against the stops as it were. And that is certainly possible
      But in fact, your description of “porpoising” into and out of chaos seems equally likely in any natural system as is the alternate condition of a stable position slightly into the turbulent realm.
      Egads … so much to learn, so much to consider. Thanks so much for your contributions, always interesting and thought-provoking.
      w.

      • Egads … so much to learn, so much to consider.

        You got that one right, good buddy.
        Consider the humble “random number generator”. There is no such thing, of course, but many of the ones that exist are pulled from chaotic maps. This parentage is not mathematically distinguishable by any simple statistical test, because if it were, the sequence produced would not be random! Or if you prefer, the whole point is to make the sequence look random in the specific sense that it will produce a uniform distribution of p-values in an ensemble of statistical tests against the null hypothesis “this is a perfect random number generator”, where success makes it indistinguishable from a truly random sequence, if such a thing has any meaning in a deterministic Universe, if the Universe is truly deterministic. Lots of ifs in there, some of them pretty fundamentally unresolvable and hence with answers on the edge of religion.
        I should also likely have qualified my remarks. It is my opinion that the Earth is a complex system — a system with a very, very high dynamic dimensionality and with “truly chaotic” dynamics visible in abundance throughout. For example, turbulent flow is chaotic. Does it make sense to assert that the climate is not chaotic in some average variable when one can observe chaotic turbulent flow in everything from the whistling of the wind to the crashing of the wave and water in the local dynamics? To put it another way, suppose one takes a limit cycle — any limit cycle, one with a normal or strange attractor — and averages it. Do you expect the average to be chaotic?
        This again is a particularly apropos question with regard to temperature in particular, since temperature is formally defined in terms of an equilibrium average in the first place, and has meaning in a non-equilibrium open system like the sun-driven coupled rotating earth-ocean-atmosphere only by generating a coarse grained local definition of temperature that is itself subject to many assumptions that are often untrue.
        To take this coarse-grained average definition of local temperature in a large system and then to superaverage it over the entire system into an “average temperature” that literally has no microscopic meaning and is not connected to any sort of sensible assertion concerning the distribution of energy within the system is itself a bit sketchy. But why one would expect to see chaotic internal dynamics within the system reflected in this average is beyond me.
        To understand my point, take (say) 100 simple iterated maps in the chaotic regime — the logistic map with R=4, for example — and start them with distinct x and/or a distinct R. Imagine that each x is a sample of local dynamics on a 10×10 grid. Average the resulting series. Is the result still distinguishably chaotic? Now try a 1000×1000 grid. Still chaotic? Now try a grid like that of a climate model, or a grid like the piss-poor examples of sampling that are used to generate an average global temperature anomaly (where we’ve added yet another step to the processing by subtracting an arbitrary number from the average itself so that we are no longer even looking at the original quantity). You are still thirty spatiotemporal orders of magnitude short of the Kolmogorov scale of local atmospheric turbulence.
        This is why I constantly bitch about the idiocy of the IPCC et al averaging over the output of many perturbed parameter ensemble results from the individual GCMs, and then superaveraging the results and calling it a “prediction” (or projection, or guess, or whatever you like) of the future climate. O. M. G. There are only a few dozen unproven or known false assumptions in that process. One reason I’ve started to look into chaos more carefully is to get some idea of just how false they are. I’ve already been able to show to my own satisfaction that systems with nonlinearities capable of producing chaos are not only potentially infinitely sensitive to initial conditions, they are infinitely sensitive to things like stepsize and integration algorithm (which is kind of obvious, but it is nice to watch it happen). Changing stepsize for any algorithm can change the character of the system in dramatic ways. Averaging can completely destroy the appearance of chaos (or make it look “merely random”), and do it quickly as you observe above.
        There is a temptation to say that climate is some sort of average over weather and hence is not chaotic. Perhaps that is true. But that does not mean that climate is predictable when the underlying weather is chaotic, because our climate today is the result of integrating over the chaotic weather over the entire interval in between, and we do not end up in a “state” that is the average of many such states, we end up in one specific state.
        rgb

    • “It is also difficult to map out a strange attractor with just one variable and time”
      Plot on one axis, “y”, on the next, “dy/dt” on the third, “d2y/dt2”, etc. It jumps right out. It’s how I make my Arctic Ice predictions a few months before the minimum.

    • Robert –
      Your comments about the oscillator simulation reminded me of the case where one is attempting a fixed-point (integer) digital sinewave oscillator where something very much like an “attractor” saves the day.
      First – an aside: A popular DSP lecture “War Story” is that at Bell Labs, the “theoreticians” sent instructions to the technicians to construct such an oscillator, but then thought better of it, and belayed the order. Of course, when the halt message arrived, the oscillator was already running. Great story of course – Details lacking. It was generally supposed that it was simply a matter of recognizing that for the oscillator to work, we seemed to need poles EXACTLY on a unit circle, almost always impossible with Cartesian coordinates. In hearing and retelling the story for dozens of years, I became curious about the details, and my colleagues told me it was Jim Kaiser’s group. So I asked Jim (then down with you at Duke!) and he told me that he had never even HEARD the story!
      The figure below shows the basics.
      http://electronotes.netfirms.com/FIXEDPOINTOSC33.jpg
      I will post the full 50 page of this 1995 report on my Electronotes site if there is any interest (email me). The two examples selected here show a NEAR nominal result (A) where the major non-ideal effect is a small amount of harmonic distortion due sample roundoff. Example E is more typical. It is a (NON-CHAOTIC) length 154 repeating sequence (originally termed a “limit cycle”) that closely resembles a sinusoidal oscillation, except the 13 cycles should have been 13x(360/30) = 156 samples. The frequency is thus a tad high (and there is a lot of harmonic distortion). A third case (not shown here) is where the sequence never returns to the initial state, but nonetheless becomes periodic by returning to SOME OTHER two states (often with quite different amplitude).
      Here the structure “finds” a sequence. Why does this non-linear time-varying deterministic device seem to behave reasonably well (although LOOK temporarily chaotic)? It seems unlikely. The simplest explanation is that eventually, it has to find two states it has previously visited because there are only a finite number of consecutive pairs of states (fixed-point arithmetic). Moreover, because of the sinusoidal “model” of the second-order network, it is primed to find a solution sooner, rather than later.

  49. rgb See comments
    http://wattsupwiththat.com/2015/10/22/is-the-climate-chaotic/#comment-2054886
    and
    http://wattsupwiththat.com/2015/10/22/is-the-climate-chaotic/#comment-2054960
    above.
    Don’t you think that ,for purposes of discussing the climate system as a whole over geological time scales, the system is obviously stable.
    Within that framework the rate of change of particular parameters may vary significantly as the combination of the absolute values of certain individual parameters crosses particular absolute thresholds.
    As an example see the rapid rise in temperature at the end of individual ice ages.
    Those living during such times might well get the impression the climate is unstable – or even chaotic !!.

    • Oh dear, some others that do not bother to find out what chaotic means. A double pendulum has also a bounded trajectory, it does not mean it is not chaotic. There are plenty of very simple systems with bounded trajectories, which are chaotic.
      One post even talks about some sort of ‘equilibrium’. Sorry, there is no equilibrium. The Earth system is not at equilibrium and it cannot be. It’s a rotating body which receives energy from the Sun while sending energy into space. That cannot reach equilibrium. Not even a dynamical one.

      • When discussing the behavior of the climate, temperature is the most useful parameter.
        MathWorld says
        “The boundary between regular and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Strogatz 1994, pp. 363-365). ”
        Using this practical criteria there is no evidence of chaotic climate behavior for the last 6 -700 million years.

      • When discussing the behavior of the climate, temperature is the most useful parameter
        ////////////////////
        Why? It would appear that you have been suckered by the warmists who have been forced to change the theory from global warming to climate change, and when they do this they only advance evidence that temperature has increased, since there appears to be little evidence that other climatic parameters have changed at all. You are playing their game.
        What about floods or drought which could be much more significant.
        What would happen to places like India if there were say 6 or 7 years without monsoon?
        We know what happened to Egypt when the Nile didn’t flood. The problem was not temperature.
        The reason we are fixated on temperature is that the radiative properties of certain gasses can absorb and re-emit photons and it is claimed that this property leads to a temperature increase. It is not a laboratory characteristic of CO2 that it causes climate change. Temperature is important for considering AGW, but not necessarily climate change. As I said in my post (October 22, 2015 at 6:22 pm0 above:
        “Far too many people equate temperature with climate. Temperature is merely one of the variables that make up climate (others include humidity, precipitation (amount, nature and cycles thereof), air pressure, wind, windiness, influenced by El Nino/La Nina conditions etc etc). Each of these parameters is variable, and constantly varying within bounds, and each may interact with each other.”
        There has been some modest warming in the 20th century, but there has not been any climate change. No region has changed its Koppen (or equivalent) characterisation.

      • richard verney,
        Correct. And I would even quibble with this:
        When discussing the behavior of the climate, temperature is the most useful parameter
        I think the temperature trend is the most important and useful parameter. Temperature is only a snapshot in time.
        Another climate parameter: is the trend accelerating? Some folks still claim that the rise in sea levels is accelerating. If so, that would be a serious cause for concern. Fortunately, they are wrong.
        And Arctic ice: is the reduction around 2006 – 2012 a trend? Or is it just a natural fluctuation?
        The ‘climate’ consists of many parameters. Those parameters have been wide ranging over the past century or so. But the current climate parameters are not unusual, and they are certainly not unprecedented. They are normal and natural; every climate parameter observed now has been exceeded in the past, repeatedly, and to a much greater degree than now.
        Therefore, what we are observing now is just the normal, natural, and cyclical changes that always occur. When Occam’s Razor is applied, we see that there is nothing to be concerned about.
        The climate Null Hypothesis has never been falsified. That means that the Alternative Hypothesis — that human CO2 emissions cause global warming and other calamities — has been falsified, because the Null Hypothesis and the Alternative Hypothesis cannot both be right at the same time. One of them must be false.
        That’s the rational, scientific view. Not many skeptics say that CO2 has zero effect. Most skeptics say that the effect of human CO2 emissions on global T is so minuscule that it can be completely disregarded for all practical and policy reasons.
        But the climate alarmists’ goal is not scientific knowledge. If it was, the question would be settled by now. The real, unstated goal of the alarmist brigade is the passage of a carbon tax, and handing over national enforcement authority to a world body.
        A carbon tax would greatly increase the cost of all goods and services, without a commensurate rise in incomes. And of course, the UN is always angling for more power. But no one in their right mind would hand the UN or anyone like it that kind of power, so they play the game of climate alarmism. That requires that they lie about it.
        That’s the elevator speech explaining what’s going on. It isn’t about science; it never was. Nor consensus. It is greed for money and power. No more and no less.

      • Richard. You say – “There has been some modest warming in the 20th century, but there has not been any climate change. ”
        Complete nonsense -the climate changes continually and the changes in global temperature are the most useful metric as a general measure of that change – Especially when the Northern and Southern hemispheres are considered separately.
        If you think I have been suckered by the warmists you have never read any of my blog-posts at
        http://climatesense-norpag.blogspot.com
        where you will find my forecasts of the coming cooling.

      • =============
        The boundary between regular and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Strogatz 1994, pp. 363-365). ”
        Using this practical criteria there is no evidence of chaotic climate behavior for the last 6 -700 million years. – Dr. Page.
        =============
        1 polar vortex x 2
        = 2 polar vortices:
        http://eoimages.gsfc.nasa.gov/images/imagerecords/2000/2832/antarctica_ept_2002267.jpg
        That was only 13 years ago.
        January last year:
        http://climatechangenationalforum.org/wp-content/uploads/2014/01/GoHomeArcticYoureDrunk.png
        The flat “average global temperature” trend since ~97 hasn’t been useful for predicting climate on the ground. Richard Verney makes a good point.
        http://www.sea-way.org/blog/From_space02.jpg

      • I am well aware of your posts, and your predictions, and we have exchanged comments upon some of your posts. Time will tell whether your predictions come to pass, and perhaps, if this is accompanied by better knowledge and understanding, time will also enable us to ascertain whether you were right as to cause.
        But with respect, Dr Page, on this particular issue (ie., whether temperature is climate or the most important proxy for climate), it is you who are speaking nonsense, and this is because of a fundamental failure to grasp what Climate is.
        Climate consists of a range of parameters, temperature being just one of many parameters that make up Climate. Materially, each of these parameters is never in stasis, and each of these parameters is constantly varying within bounds (some of which can be quite wide). This means that change of any one parameter is not Climate change, nor even evidence of Climate change. Change is just what Climate is. It is only when a parameter or a number of parameters varies beyond the bounds of natural variation AND stays there for a protracted period of time, is there evidence of Climate change, and depending upon the extent to which the bound has been broken and the length of time that that variable has exceeded its natural bound will there be Climate change.
        If I look at the UK weather forecasts over the next week or so, I will find temperature variations of up to 7 or so degrees. Consistently, I will find areas of Scotland being 1 to 2 degrees lower than other areas of Scotland, Scotland generally being 1 to 2 degrees lower than the boarders, the boarders being 1 to 2 degrees lower than Wales, Wales being 1 to 2 degrees lower than the Midlands, the Midlands being 1 to 2 degrees lower than the South West, the South West being 1 to 2 degrees lower than the South East, the South East being 1 to 2 degrees lower than the Channel Islands, Ireland and the Scottish Isles will also have different temperatures. One could divide the UK into at least 7 may be 10 different temperature zones, but no one has ever claimed that the UK has 7 to 10 different Climates.
        The fact that there are variations in temperature of this order over what is a very small country just confirms that temperature does not alone dictate Climate. Climate is regional (not global), and even on a regional basis there is much variation in it, even to the extent that you get micro climates for example in the shadow of mountains, on the coast, particularly if warm currents pool, or if you have mountains and coast in close proximity etc. etc
        In the UK, it is thought that the 1530s/1540s were the warmest decade on record. Did the UK have a different climate then? I would say no, it was simply that it was slightly warmer.
        Climate cannot be measured over 30 years, or even a 100 years. If one goes back through the Climate history of a country say for a 1000 years then you get a good insight into what the Climate of that country is. and the extent of the upper and lower bounds of temperature, rainfall, snow fall, floods, droughts, storminess etc etc.
        There is no evidence of actual Climate change, albeit that there is some evidence that there has been some warming since the depths of the little ice age with that warming being variable in fits and starts, and not correlating with the rise in CO2. Whether it correlates with anything else is moot, and presently we do not understand why there was the Holocene Optimum, the Minoan Warm Period, the Roman Warm Period, the Medieval Warm Period, the Little Ice Age, the late 20th century warming, the current pause. All we can say is that CO2 does not correlate with such events, and that it appears that the planet is on a downward trend which will be interrupted in fits and starts, heading back into the throws of the current ice age. ..
        I know that you are not a warmist, and I shall continue to read all your posts and continue to watch to see whether your predictions come to pass.

    • Mr. Page,
      The Webster says;
      Chaos;
      complete confusion and disorder : a state in which behavior and events are not controlled by anything.
      You win this debate, I say. No contest.

  50. Temperature would be the last metric I would look at for this issue. It is the end result of a mix of factors, of which, different mixes can result in the same temperature outcome. These other factors are themselves a result of topography and regional climate made up of temperature range, humidity, sun days, cloudiness and cloud patterns, etc.). Wickedly complex question. Wickedly complex investigation.

    • Pamela
      The question is so wickedly complex that it is inherently insoluble by numerical modeling methods so that another forecasting approach is necessary. I will repeat part of an earlier comment.8:41 AM above,
      “This does not mean, however, that climate can be forecast using reductionist climate models.The modelling approach is inherently of no value for predicting future temperature with any calculable certainty because of the difficulty of specifying the initial conditions of a sufficiently fine grained spatio-temporal grid of a large number of variables with sufficient precision prior to multiple iterations. For a complete discussion of this see
      Essex: https://www.youtube.com/watch?v=hvhipLNeda4
      Earth’s climate is the result of resonances and beats between various quasi-cyclic processes of varying wavelengths combined with endogenous secular earth processes such as, for example the changing geomagnetic field.. It is not possible to forecast the future unless we have a good understanding of where we are at present time in relation to the current phases of these different interacting natural quasi-periodicities which fall into two main categories.
      a) The orbital long wave Milankovitch eccentricity,obliquity and precessional cycles which are modulated by
      b) Solar “activity” cycles with possibly multi-millennial, millennial, centennial and decadal time scales.
      The convolution of the a and b drivers is mediated through the great oceanic current and atmospheric pressure systems to produce the earth’s climate and weather.
      After establishing where we are relative to the long wave periodicities , especially the solar millennial cycle, to help forecast decadal and annual changes, we can then look at where earth is in time relative to the periodicities of the PDO, AMO and NAO indices and based on past patterns make reasonable forecasts for future decadal periods.
      For the latest forecasts and complete discussion see
      http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html

    • Surely, enthalpy is more important a metric, but it seems to be harder to get a hold of. If the earth was a glass of water full of ice cubes a lot “warming” would need to go on before temperature began to rise. Temperature is a kind of ‘remainder’ of all the other forms that the sun’s energy contributes to the earth and is therefore very incomplete a measure of how much ‘heat’ is retained.
      I am convinced that we best study the polar regions for our indications of a warming planet – polar amplification serves as a kind of vernier for examining essential climate change. If the Arctic ice recovers, – it certainly doesn’t seem to be getting worse- then all alarm is empty noise. To bad we didn’t do more study in the polar regions during the 1930s. We probably would be able to say that today’s ice extent is nothing to worry about. Regarding the temperature in the tropics – I recall the same temperatures in Lagos in the mid 1960s to what I see now. Let’s stop worrying about temperature change in the tropics – it aint happening there and even CAGW theory says it shouldn’t

  51. “A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.”
    Um. No. You’ve linked to a post on the lorenz attractor. Ever put in the initial state (0,0,0)? It neither converges nor diverges. A chaotic system, by definition, is a system with a non-integral dimension, a fractional dimensionality. The Lorenz attractor, for instance, has a Hausdorff dimension of 2.06, according to wiki https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension . It covers more than an area. It fills less than a volume. It has a fractional dimension, ergo it is chaotic.
    “Whether a dataset is chaotic or not is generally assessed by looking at the Lyapunov exponent.”
    No, it’s not. One assesses it by looking at the dimensionality. The Lyapunov exponent (singular, when did it become singular?) is not generally used. That’s why wiki has half a hundred Hausdorff dimensions listed for various attractors…. and no list of Lyapunov exponents for various attractors, AFAIK. Mandelbrot originally used a Hausdorff dimension to describe his fractal. He did not use a Lyapunov exponent or even more than one Lyapunov exponent (I love the fact that you stated it as a singular. It makes me chuckle.). Professionals use Hausdorff dimensions. Authors like James Gleick use Lyapunov exponent(s).
    (Aside #1: I just read this in wiki, “Thus, there is a spectrum of Lyapunov exponents— equal in number to the dimensionality of the phase space.” Didn’t you at least read wiki? Didn’t that clue you into the idea that there was more going on than a single exponent? Didn’t the phrase “dimensionality of the phase space” stop you short? Didn’t you wonder what wiki meant by “dimentionality” or “phase space”? Do you even know what “phase space” might be?
    “An example is the ocean surface. If you drop two sealed empty bottles overboard in mid-ocean, one on each side of a boat, they will drift apart over time. This separation will be slow at first, and then faster and faster as the two bottles encounter different winds and currents in different areas.”
    This is an assertion. There is no evidence that “then faster and faster” will happen. That’s an assertion. You have not differentiated between chaos and two bottles on a random walk. Hint, if the two bottles can hit each other then there is randomness involved. They can hit each other. Again, this would be impossible if they were in a system dominated by chaos.
    (Aside #2: I should turn this aside into another post… Didn’t the sentence from wiki “A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness)” bring you up short? Did a word like “compactness” prompt you to think why two bottles in the ocean might not be a good example of chaos and a better example of randomness….. No? NO?)
    “Now, these results were a great surprise to me.”
    Oh we know. Physics and math is generally a surprise to you. You began your post stating as a fact that the ocean is chaotic and not random. You then do some plots and find that the systems you are interested in are dominated by randomness. It never occurs to you to go back and figure out that the two bottles you used to explain chaos are sitting there hitting each other over and over which would be impossible if the ocean were dominated by chaos. Once again you’ve made a hash of things.
    That result not a great surprise to me.
    Once again Willis I ask that you stop doing this. You are consistently misinforming people.

      • 1) Of what relevance is fractal geometry to the question of chaos?
        Not geometry per se but a fractional dimension has everything to do with chaos. That’s why everyone gets so interested in Mandelbrot’s work. Because he showed a real-world thingy that had a fractional dimension <2, i.e. more than a line and less than an area. That is why everyone gets so excited but the Lorenz attractor. Because it has a fractional dimention <3, i.e. more than an area and less than a volume.
        Take Lorenz's equations for instance (btw- I'm about to create a conceptual example from equations…. I hate doing that.) as the state moves around, in a 2-D (area), it seems to have the same 2-D state but goes off in a different direction. So 2-D isn't enough. As the state moves around in a 3-D volume one sees missing holes between the states that are never, ever filled. 3-D is too many dimensions.
        A random variable would end up filling that 3-D volume. Eventually Willis's bottle in the ocean passes through the previous location of the other bottle. Eventually, they share exactly the same spot. Then even though the two bottles had the exact same location, they randomly walk away from each other. The bottles do not move chaotically but randomly.
        Here is a nice picture from wiki showing the random walk of a large object being pushed around by other smaller objects. https://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Brownian_motion_large.gif/220px-Brownian_motion_large.gif Think of it as Willis's bottle.

  52. Here are a few questions I don’t know the answers to but are always of interest?
    1. If the ‘global temperature index’ (whichever one you choose) is NOT chaotic, is it still possible that useful predictions cannot be made in any better way than applying the equivalent of quantum mechanical frameworks to say: ‘there are a number of allowable states (aka future temperature evolutions) which can emerge from this situation, and all we can say is that there are statistical likelihoods assigned to those potential allowable states (whose error bars may nonetheless be rather large)?
    2. Do climate indicators like ‘global temperature’ show features similar to what you see on trading exchanges, where ‘resistance points’, ‘break-out events’, ‘singularities’ occur etc etc? Singularities would, in this case, be analogous to e.g. transitions between inter-glacials and ice ages. Resistance points might be natural boundaries of variability within an inter-glacial, within a centennial or inter-centennial set up or even within PDO cycles, Hale Cycles or Sunspot cycles. I”m not sure the Quants on Wall Street will wish to be poorly paid to transfer their knowledge from algorithms to predict gold futures to algorithms to predict climate regimes, however……..
    3. If temperature indices are not ‘chaotic’, does that imply that apparently stochastic input variables like e.g. major volcanic eruptions, solar events like major flares etc etc are not, in fact, stochastic but, with sufficient scientific research and data collection/analysis, will become amenable to reasonably accurate predictions?
    4. Are ‘global temperature indices’ actually much use in predicting local/regional weather patterns (since actually what is important on an annual basis are things like rainfall patterns, temperature patterns etc not across the globe but in e.g. the Mid West Corn Belt, the SE of the UK, the Corn Belts of Ukraine and southern Russia etc etc)?
    The big questions are, I think, whether useful predictions on a 9 – 12 month scale are intrinsically possible (my hunch is they are, since I managed to zone in for the best part of a decade on likely evolution of snowfall patterns in the European Alps when I was a mountain goat for a decade back in the day, successfully predicting times of the season to ski in different parts of the alps, at least before a shift in the PDO came along), given the ability to understand and measure the right input variables, and, if so, down to what geographical range (probably easier to make average predictions for the whole of the US corn belt than it is to predict precise patterns for East Anglia in the UK, after all).
    I was most interested in the analyses a year or so back comparing the climate system to an electronic circuit – it seemed to me to be one of the best analogies.
    What is perhaps a more interesting theoretical question is this: ‘how much previous data do you need to input into your system before an apparently chaotic one becomes a non-chaotic one? Or vice versa (if a singularity comes along)?’
    It would provide scientific robustness in answering the question of how long scientists need to keep on collecting primary data before rigorous modelling is likely to bear fruit……..

  53. Dr Norman Page
    I have addressed your comment (October 23, 2015 at 9:08 pm) above at (richard verney October 24, 2015 at 2:33 am). dbstealey at (October 23, 2015 at 5:16 pm ) makes the valid point:
    “The ‘climate’ consists of many parameters. Those parameters have been wide ranging over the past century or so. But the current climate parameters are not unusual, and they are certainly not unprecedented. They are normal and natural; every climate parameter observed now has been exceeded in the past, repeatedly, and to a much greater degree than now.”
    What he is describing is just what Climate is. You say that “the climate changes continually”. I would say continuous change is simply what Climate is, such that change of any one parameter is not in and of itself Climate change. As I say change (by which I mean change within bounds as described by dbstealey) is not Climate change, it is only once the natural bounds of variability have been exceeded for a protracted period of time that one may be witnessing Climate change. That may be subtle, but it is important.
    In your comment (October 23, 2015 at 1:54 pm) you ask: “Don’t you think that ,for purposes of discussing the climate system as a whole over geological time scales, the system is obviously stable.”?
    I find that difficult to answer since it depends upon what you mean by stable and the precise context in which that is being considered. Over the geological timescale you mention (the past 700 million years or so), the Climate has varied over relatively narrow bounds bearing in mind the possible extremes that may have been available. That is probably the consequence of the fact that responses are dampened and with a large buffer because the planet is a water world with most of that water being on the surface.
    So personally, I would not say that the Climate was stable, but rather that it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics

    • Richard You say” So personally, “……. it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics” I agree – Norman.

      • Richard Better posted this way
        You say ” it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics” I agree – Norman.

  54. This conclusion:

    “however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest.”

    is true.
    It is also trivial if the question is Is Earth’s climate system “chaotic”? [in the strict sense of Chaos Theory/Non-linear Dynamical Systems].
    That the observational datasets exhibit strong stability over time is evidence that the climate system is chaotic.
    When forcings are within certain values for a non-linear dynamic system, it forces first stability, then as forcing changes (usually increasing, but can be decreasing), one begins to see periodic results, rapidly doubling and re-doubling, leading to “chaotic” results.

    “In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system.”

    https://wattsupwiththat.files.wordpress.com/2015/10/bifruct-annot.jpg
    Stability in spite of perturbation, the rapid return to the stable value after perturbation, is a indicator that we are dealing with Chaos Theory’s “strange attractors” (later on, when things gets wild the attractors can indeed be strange) — kick a value out of whack with a decade of volcanism, for example, and it rapidly (relatively) returns to the “preferred” value range (the “attractor” of the system).
    https://wattsupwiththat.files.wordpress.com/2015/10/stability_at_r__28_650.jpg
    In the case above, the value had been programmatically perturbed way out of the stable range. Within a dozen or so iterations (days, years) it has returned to the point of stability. This type of behavior is what I (and many others) see in Earth’s climate today, “those six observational datasets” can only be perturbed short-term and are, in the Chaos Theory sense, “attracted” back to their point of stability given the existing range of forcings.
    In due time, I will post my Chaos Part II essay, which is on this particular point.

    • Mr, Hansen,
      “That the observational datasets exhibit strong stability over time is evidence that the climate system is chaotic.”
      Any chance the stat/math folks have gone and picked the wrong term? Strong stability over time is pretty much the opposite of chaotic, in English, ya know?

      • If a biologist said; That the observational datasets exhibit strong stability over time is evidence that the living system is chaotic . . wouldn’t you question the wisdom of using that term in particular, to describe this sort of . . effectual stability over time? Or I said; That catcher sure is good with wild pitches, the runners rarely have a chance to advance, his ball handling systems are chaotic . . would those be “technically” correct in the sense your sentence was?
        (If so, I’m just sayin’ ; )

      • Reply to JohnKnight ==> First you have to have the right definition of Chaotic — you must know and use the term in its own frame of reference, which is Chaos Theory — w’s essay, and all these comments, are based on the premise that we are talking Chaos Theory when ever the words “chaos” and “chaotic” are used. This idea stems from the IPCC’s early admission about the Earth’s Climate System: “The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.” TAR – Working Group I: The Scientific Basis
        If you are not passingly familiar with chaos and chaotic in that sense, the Wiki has a nice primer .
        In comments to my first essay on Chaos and Climate, at least one engineer stated that engineers depend on this stability in non-linear dynamic systems (which are, in the vast majority of cases, chaotic) to hold things together and are very careful to avoid parameters which would push their systems past the known points of stability. Engineers love this feature, as if something twangs the bridge, say, the bridge ‘wants’ to get back to stability.
        If after reading the Wiki primer and following some of the links, if you still have questions, ask me.

    • Kip Hansen,
      Thanks . . and yeah, I get the idea, it’s just a very poor choice of term to my mind. Complex order is not chaos . . it’s reality ; )

      • Reply to JK ==> It is worse than you think….
        Chaos Theory is not a theory, there is no theory.
        Chaos Theory refers to systems that are strictly, thoroughly deterministic but not at all predictable once they enter the chaotic realm.
        Chaos Theory predicts the stability I mention above.
        The Chaos comes from a deterministic system, like a simple formula, whose results are easily determined, but only one step at a time, it is impossible to know what the result ten steps ahead will be without going through the intermediate nine steps. Edward Lorenz found that one could fairly accurately predict tomorrow’s weather given enough correct information about today’s weather…but not next weeks (except in a general way) — results are very sensitive to initial conditions, the tiny changes in initial input may cause big changes in output down the line (and the may is important, they may not do so).
        Fun stuff — an introductory understanding of Chaos Theory can change one’s appreciation of the physical and mathematical world.
        .

      • Kip.,
        I don’t doubt that an introductory understanding of so called Chaos Theory can change one’s appreciation of the physical and mathematical world . . but it would help the whole appreciation drive if mathematicians would appreciate the language realms a bit more, and avoid using terms for their ideas/concepts that generate contra-logical language snafus when the ideas are spoken of outside the shop, so to speak.
        I’ve given it some thought (yer welcome math/stat heads ; ) and a bit of word searching, and I suggest ‘chaosh’.

  55. Our largely water and less so moving land masses planet with its highly variable atmosphere is the source of both long term and jagged temperature trends between the extremes of the various forms of Milankovitch cycles. I can well imagine an orbital tilt of one variety or another causing changes in our semi-permanent and permanent oceanic/atmospheric systems, along with the way discharge and recharge regimes work.
    http://www.pnas.org/content/112/12/E1406.full.pdf

  56. M Courtney October 22, 2015 at 2:58 pm

    Yes. And that is what ristvan implies.
    That the weather has a strange attractor that makes temperature, air pressure, cloud cover (etc.) consistent in the long term.

    Thanks, M. I wrote this post to engender discussion and encourage questions.
    Rather than saying climate has a “strange attractor”, I’d say that the climate has a number of emergent thermoregulatory phenomena operating that keep those variables consistent.
    Which brings up an interesting question:
    Is a thermostatically controlled system chaotic?
    I’d say “Could be, but not necessarily so”. I’d also say that if the thermoregulatory phenomena are independently mobile (e.g. dust devils, thunderstorms) that chaoticity would be almost assured. But like I said … what do I know, I was born yesterday.
    w.

  57. Dinostratus October 23, 2015 at 11:26 pm

    Willis conflates random with chaotic in his ocean bottle example. They are different.

    Indeed, random and chaotic are different. I showed the difference in Figure 1.
    As to the ocean being “random”, I’d disagree. The ocean has tides, resonances, whirlpools, currents, bifurcations, tide rips, and waves from the surface down to the bottom. While you could describe it as “chaotic” or “turbulent”, I don’t think “random” even begins to touch the nature of the ocean.

    A random variable would end up filling that 3-D volume. Eventually Willis’s bottle in the ocean passes through the previous location of the other bottle. Eventually, they share exactly the same spot.

    Not true in the slightest. For example, the bottles will never go to certain areas, because the areas are upwelling and thus the current goes outwards in all directions. As an example, the Argo floats rarely sample parts of the normal location of the ITCZ, because of the upwelling there. So no, the movement of a floating bottle does NOT end up filling that 2D space, and thus, a bottle doesn’t walk randomly about the surface of the ocean.
    w.

    • Thank you for being stubborn and unwilling to learn from your betters. I was concerned that you might figure something out then start to misinform and confuse people at a higher level.

      • Dino, normally a scientific discussion starts with someone making a scientific claim. Then someone else disagrees. Then the first person comes back and defends their view, pointing out errors and providing support for their view. Then the second person does the same, clarifying and explaining. It is an interchange of ideas.
        With you, on the other hand, you make a scientific claim. Then I disagree. And your response?
        Rather than defending your claim, you jump right to the easy insult, that I’m “unwilling to learn from my betters”.
        Dino, I learn from my betters all the time … but you are definitely not among them. I suspect you could be, but instead you spend all of your time attacking me instead of showing what’s wrong with my ideas.
        Which is why I usually just let your attacks float on by. But in this case, I had hope, because you actually and most unusually made a scientific comment instead of your usual meaningless, uncited attack on my abilities, my style, or my knowledge.
        But that didn’t last long. One comment from me on your scientific claims and you abandon them with no defense, instead you’re back to slinging mud …
        Sorry, son, not impressed. Come back when you’ve grown up and you wish to discuss other peoples’ ideas and not their learning styles …
        w.

  58. The future climate is not chaotic:
    It is always getting hotter, with 105% certainty, er, uh, I mean 95% certainty!
    The past climate is very chaotic:
    It is changing every year, sometimes more than once a year, and as a result of these adjustments, re-adjustments, and re-re-re-adjustments, the historical average temperature data are in a permanent state of chaos .. or perhaps I should say in limbo.

  59. So many half thoughts. When you aren’t specific you come across as someone with a thesaurus on his knees.

  60. My seat of the pants thinking is this: There is a millennial scale cycle that rules thusly: based on orbital mechanics solar insolation shifts also shifts the ITCZ important global oceanic-atmospheric teleconnected systems driving whether or not there is a continued net disgorgement of heat from the oceans (which occurs rapidly), or a jagged recharge which results in the typical downward steps observed in the paleo-reconstruction due to the continued recharging of ocean heat, leaving us without benefit.
    In terms of what puny humans experience in our life times, it sure feels chaotic. But over the course of 10’s of 1000’s of years, it is anything but chaotic. The metronome of Earth’s orbital mechanics makes it non-chaotic.
    http://www.climategeology.ethz.ch/publications/2014_Schneider_et_al.pdf

  61. Earth’s climate system is random and chaotic is more correct then saying the earth’s climate is stable.
    Chaotic is in the eyes of the beholder. The earth’s climate having the ability to go from a glacial state to a glacial state and have abrupt climate changes at times satisfies chaotic for my way of thinking.
    A non chaotic climate would be a global temperature changed over 1000’s of years that varied by less then 1C which we know is not the case although the overall global temperature change from glacial to inter glacial periods is perhaps ONLY between 4-7C.
    That being deceiving due to the fact that the tropics relatively small temperature changes from glacial versus inter- glacial periods of time tempers the global total temperature change.
    In closing a measure of how chaotic the climate is should be what impact does it have upon humanity and from this approach I would say the climate changes sufficiently enough to have a very big impact on humanity which makes the climate chaotic regardless.

  62. The climate is random due to the fact it is subject to random extra terrestrial events and random terrestrial events.
    It is chaotic because the climate is always a reflection of everything that is effecting it at a given time and all the forces that effect the climate at a given time are never the same ranging from land/ocean arrangements, to solar variability , to Milankovitch Cycles, geo magnetic field strength, lunar influences, the ice dynamic etc etc.

    • Reply to S.D.P. ==> It seems that you are using the standard English definitions of chaotic and random.
      Chaotic, in the sense used throughout this essay’ is:

      The IPCC’s early admission about the Earth’s Climate System: “The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.” TAR – Working Group I: The Scientific Basis

      If you are not passingly familiar with ‘chaos’ and ‘chaotic’ in that sense, in the Chaos Theory sense, the Wiki has a nice primer.
      There are a half dozen absolutely indispensable books on Chaos Theory. If you haven’t already, read James Gleick, Edward Lorenz, and Ian Stewart for a quick start.

      • I think it would do some good to define dimensionality and show people the difference between chaos and randomness. WIllis has made a hash of things and has really confused people by deciding to use “the Lyapunov Exponent” as opposed to the dimension.

      • Reply to Dinostratus ==> Edward Lorenz (the “Father of Chaos Theory”, a rather incorrectly assigned title, which he freely admits, but is stuck with) defined chaotic systems as those that can be shown to be

        “sensitively dependent on initial conditions” with “behavior that is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight amount of randomness, but h. This means that the current state completely or almost completely determines the future, but does not appear to do so.” Lorenz goes on “sensitive dependence [on initial conditions] can serve as an acceptable definition of chaos, and it is the one that I shall chose.”
        The Essence of Chaos Chapter 1 Pg 8

        Lorenz explores Fractality as a separate, related, concept, in the same book, in Chapter 5 “What Else is Chaos?”. It was Mandelbrot, however, that “coined the word fractal to describe systems with a fractional dimensionality”.
        Lorenz’s definition above gives at least a conceptual difference between randomness and “chaotic”. In random systems, the next state is not deterministic, it could be any of a large or even infinite number of possibilities. In a “chaotic” system, the next state is entirely (or nearly so in the real world) determined by the present state, it can be ONLY ONE value. The catch is, it is not possible to determine what that value will be until one gets to it, it can not be predicted, which is why chaotic systems (in the chaotic state) “do… not look deterministic.”
        I have discussed above that non-linear dynamic systems, which by their nature are chaotic, have periods of stubborn stability and odd, beautiful order (strange attractors), snapping from one to the other.
        I do agree that w’s choice of “the Lyapunov Exponent” was ill-though out — possibly driven by the finding of a R-function “the lyap_k function from the R package tseriesChaos”.
        The choice of datasets seems to be itself random or opportunistic (searching for one’s car keys under the lamp post because that’s where the light is good).
        It is virtually impossible that “HadCRUT4 Monthly Mean Surface Air Temperatures 1850-2015” could be chaotic — it is not itself “real”, being an average of averages of averages (ad infinitum) of ranges of daily temperature readings at random locations — any real meaning smoothed, homogenized, kriged and finally politically adjusted far past any hope of representing actuality. One would not expect “temperature” to be chaotic in any case. (see Wyatt). Two datasets of a randomly chosen (?) river in North Africa reflecting regional rainfall at its headwaters. Tides in a Scandinavian sea port (climate related?) and finally two 100-year-plus datasets of temperature, measured to +/- 0.5° C, of essentially the same global place (Ireland and Central England) — surface air temperatures!
        Made a hash of things and indeed.
        In w’s defense, he just plain seems to have a misunderstanding of Chaos Theory and its implications from the git-go. His “two bottles in the ocean” example shows how “off” his concept of Chaos is.

      • Oops — something nipped from the first blockquote:

        “behavior that is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight amount of randomness, but does not look deterministic.. This means that the present state completely or almost completely determines the future, but does not appear to do so.”

      • Kip Hansen October 26, 2015 at 7:34 am

        The choice of datasets seems to be itself random or opportunistic (searching for one’s car keys under the lamp post because that’s where the light is good).
        It is virtually impossible that “HadCRUT4 Monthly Mean Surface Air Temperatures 1850-2015” could be chaotic — it is not itself “real”, being an average of averages of averages (ad infinitum) of ranges of daily temperature readings at random locations — any real meaning smoothed, homogenized, kriged and finally politically adjusted far past any hope of representing actuality. One would not expect “temperature” to be chaotic in any case. (see Wyatt). Two datasets of a randomly chosen (?) river in North Africa reflecting regional rainfall at its headwaters. Tides in a Scandinavian sea port (climate related?) and finally two 100-year-plus datasets of temperature, measured to +/- 0.5° C, of essentially the same global place (Ireland and Central England) — surface air temperatures!

        Gotta tell you, Kip, you joining Dinostratus in the “I’ll tell you what’s wrong … but not what’s right” club is pathetic. You’re both more than happy to tell me that I’m wrong, wrong, wrong, but you are surprisingly coy and reticent about what’s right.
        Take your comment above. I chose those datasets because they were a widely disparate group, and I wanted a wide sample of possibilities. You whine that HadCRUT4 surface air temperature couldn’t be chaotic because it’s an average … but then I knew that it was an average, which is why I also included a long single-station dataset, to see if the averaging made a difference. You claim the single-station dataset is wrong, wrong, wrong … but you don’t say one word about why.
        More to the point, however, if you are such a freakin’ expert on what datasets I SHOULD NOT have used, where is your expert opinion about what dataset I SHOULD have used?
        And if you think I’m not using the lyap_k function correctly (despite the fact that I used it to clearly distinguish between chaotic and non-chaotic datasets), then how about YOU SHOW US HOW TO USE IT CORRECTLY!
        Because between you and Dino, your endless whining and negativity is getting quite boring. You need to either lead, follow, or get out of the way, because at present neither of you are contributing anything to move the conversation forwards. I may indeed be very wrong … but until either of you can demonstrate what is right, you’ll get no traction with me.
        w.

      • Reply to willis ==>
        Your posited defintion:

        …what is meant by “chaotic”? A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.

        needs a reference ….
        Your exercise did find what it found — the data sets tested with a specified R-function showed no signs of being chaotic. That’s your finding. (and I, for one, and rgbatduke, for another, would not have expected anything different.) That says nothing whatever about Earth’s Climate System.
        I have given above Lorenz’s definition of Chaotic Systems — which you may not like or agree with, but it certainly is well accepted in the field.
        Rather than get into some kind of back-and-forth with you, I’ll just say that I agree with Dr. Brown [rgbatduke] here and here above. Quoting excerpts from him at the links provided:

        “So you are looking at the wrong thing, in the wrong dimensionality.”
        “Does it make sense to assert that the climate is not chaotic in some average variable when one can observe chaotic turbulent flow in everything from the whistling of the wind to the crashing of the wave and water in the local dynamics? To put it another way, suppose one takes a limit cycle — any limit cycle, one with a normal or strange attractor — and averages it. Do you expect the average to be chaotic?”
        “But why one would expect to see chaotic internal dynamics within the system reflected in this average is beyond me.”

        All but one of your data-sets are averages or means — most averages of averages of averages — in which one could not rationally expect to find signs of chaos even if it were in the specific system the data-set pretends or is proffered to represent. Looking for chaotic dynamics in such is beyond me too.
        CET Record Daily Maximum Temperatures 1878-2015 is at least a single data point set — but one would not expect temperature — even Maximum Daily Temperature — to be Chaotic….for too many reasons to go into here…but Dr Brown’s comment apply…”So you are looking at the wrong thing, in the wrong dimensionality.” or read Lorenz, “The Essence of Chaos“, he discusses this point.
        I accept that you never considered the Stockholm Monthly Tides 1801-2001 to be a climate indicator — but I point out that it too is an average of averages of averages…..in a system (oceanic tides) not generally considered to be Chaotic.
        It is the Earth’s Climate System itself that is chaotic in nature — two major coupled yet independent chaotic dynamic systems — the atmosphere and the oceans — in which the chaotic nature is evident in everyday life, without resort to “statistical packages”.
        It is simply my hope to help some others here better understand the topic of “Is the Earth’s Climate System ‘chaotic’?” a little more clearly.
        If you feel inclined to argue (in your usual inimitable style), please argue with Dr. Brown — whatever you say to him will spur him on to posting even more helpful, educational comments here — always welcome.

      • Kip, thank you for your reply. However, perhaps you didn’t notice that a) I asked three clear questions, and b) you did not answer even one of them.
        I didn’t ask for another lecture on how foolish I am. I didn’t ask what I was doing wrong, which you and Dino seem obsessed with. I didn’t ask you to refer me to Dr. Brown.
        Here’s what I asked:

        You whine that HadCRUT4 surface air temperature couldn’t be chaotic because it’s an average … but then I knew that it was an average, which is why I also included a long single-station dataset, to see if the averaging made a difference. You claim the single-station dataset is wrong, wrong, wrong … but you don’t say one word about why.
        More to the point, however, if you are such a freakin’ expert on what datasets I SHOULD NOT have used, where is your expert opinion about what dataset I SHOULD have used?
        And if you think I’m not using the lyap_k function correctly (despite the fact that I used it to clearly distinguish between chaotic and non-chaotic datasets), then how about YOU SHOW US HOW TO USE IT CORRECTLY!

        It’s great for you to wave your hands and say:

        It is the Earth’s Climate System itself that is chaotic in nature — two major coupled yet independent chaotic dynamic systems — the atmosphere and the oceans — in which the chaotic nature is evident in everyday life, without resort to “statistical packages”.

        and guess what … I agree with you.
        But without EVIDENCE our beliefs mean nothing. And this evidence can only come from what you demeaningly refer to as “statistical packages”. I believe, as you do, that the weather is chaotic … and so I do what any good scientist does, I’m looking for evidence to back that up. And the fact that you opine similarly means nothing in that quest.
        So how about you give up telling me I’m doing everything WRONG, and you pull out the dataset and the method that you think is RIGHT. In other words, don’t tell me I’m wrong, demonstrate the correct method that shows the climate is in fact chaotic.
        Because I think it is chaotic, and I’m looking for evidence that it is … and you’re not helping.
        Best regards,
        w.

      • Note to Wiliis ==> You can not possibly think that any sane person, or even a person like me, would actually engage in conversation with someone who speaks to them like this and this.
        But, I’ll trade you — fair’s fair — you respond to this:

        Your posited definition:

        …what is meant by “chaotic”? A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.

        needs a reference ….

        …..a reference that gives that as a working definition of a Chaotic System.
        If you do that, then I will answer one, simple, civilly put question (not a demand that I solve your conundrums or misunderstandings about Chaos Theory) — just one simple question.

      • In Willis’s defense James Gleick spends a lot of time going over Lyapunov Exponents over an over as part of the narrative. It makes for a good narrative. It is not a good way to quantify chaos vs. randomness but Willis’s understanding is not deep enough to know that.
        So here is an interesting numerical experiment. Take the Lorenz equations https://en.wikipedia.org/wiki/Lorenz_system and add a random permutation to one of the variables, say X. So instead of X_i+1 = f(X_i) let X_i+1 = f(X)+A*n(0,1) where n(0,1) is the unit normal distribution. How big does “A” have to be to turn X from a chaotic variable to a random variable? (Ans. “not very”) That gets to Lorenz’s point about “slight”. What is slight? What happens at the boarder of “slight” and “not so slight”?

      • Reply to Dinostratus ==> Its not that the Lyapunov Exponent has nothing to do with Chaos, which you already understand — it just can not be used as an easy-peasy test of a two-axis, time series data-set for Chaos….even if the time series data-set was appropriate.
        Lorenz speaks to the Lyapunov Exponent as well — but quite intentionally chooses sensitivity to initial conditions as his starting-point definition.
        Interesting experiment….I have played (for more hours than I care to admit, back in the 1980s) with “slight” variations of all the classic Chaos formulas…back when it used to take 30 minutes to get a clear idea how an image would develop….running Basic programs on cobbed up Commodore clones, eight inch floppies, etc. What a hoot!

      • “an easy-peasy test of a two-axis”
        I know. That’s what makes me chuckle when he brings up “the” Lyapunov exponent.
        In any event, ya, I too learned to program doing cool plots. I learned SAS and C just to do mulit-color plots on a Tektronix terminal one pixel at a time.

      • Reply to Dinostratus ==> [I have no idea if this reply will appear in the right place — the sub-thread is getting so long…:-) ]
        “one pixel at a time” — gotta love it. I was thrilled when I could do color!
        I dredged up some of my early Basic chaos stuff a couple of years ago and found that the code processed even the Mandelbrot so quickly, it was no different than opening a standard .jpg. Had to go into the code and throw in delays to get the desired effect of the set building itself in an interesting manner.

  63. Kip Hansen October 26, 2015 at 4:43 pm

    Note to Wiliis ==> You can not possibly think that any sane person, or even a person like me, would actually engage in conversation with someone who speaks to them like this and this.

    Curiously, Kip, I was just pondering the same question. Why should I engage in a conversation with people who will only tell me I’m doing it wrong, over and over … but refuse to tell me how to do it right? Where’s the cheese at the end of that maze? I’m willing to learn from anyone, but there’s no gain in such a conversation unless you are willing to show us just exactly how to do it right.

    You’ve been telling me what’s wrong but you have refused to demonstrate what’s right.

    But, I’ll trade you — fair’s fair — you respond to this:

    Your posited definition:

    …what is meant by “chaotic”? A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.

    needs a reference ….

    …..a reference that gives that as a working definition of a Chaotic System.

    Glad to. Here. Or you could try this one, which says:

    Neighbouring trajectories in a chaotic flow are found to diverge exponentially, while in a regular flow they might diverge only linearly, if at all.

    In addition, they can converge exponentially, as in the Rossler chaotic dataset shown in the figure above.
    You go on to say:

    If you do that, then I will answer one, simple, civilly put question (not a demand that I solve your conundrums or misunderstandings about Chaos Theory) — just one simple question.

    Here’s my one question.
    Are you willing to stop endlessly claiming I”m wrong, and instead demonstrate to us exactly how to do it right?
    That is to say, if you think I’m using the lyap_k function incorrectly, then show us how to use it correctly. Or if you think I’m using the wrong datasets and methods to see if the climate is chaotic, then point out the correct datasets and the right methods and post your data and code.
    After all, that’s what I do. I don’t waste time claiming people are wrong. If I think they’re doing it wrong, I don’t just say wrong, wrong, wrong. Instead, I use the datasets and methods that I think are the correct ones, and I DEMONSTRATE how I think it should be done.
    So my one civilly put question is … are you willing to do the same?
    w.

    • “Are you willing to stop endlessly claiming I”m wrong, and instead demonstrate to us exactly how to do it right?”
      Willis, there are hundreds upon hundreds of papers that show how to do it. We aren’t in charge of your literature search. We aren’t. No, seriously, we aren’t. I’ve reluctantly given dozens of hints. Reluctant because I’m concerned that you might figure something out and confuse people in new ways.
      The very first step in all good research is a lit search. Why? Because it is the most efficient way to learn and there is no joy for most in reinventing the wheel. Finding out one could have skipped months of work by just reading one more paper is a very frustrating thing. It happens to me more than I’d like to admit. Why you insist on doing even the smallest things yourself is beyond me. I suspect you’re egotistical a level of near incapacitation..

      • Dinostratus October 26, 2015 at 10:29 pm Edit

        “Are you willing to stop endlessly claiming I”m wrong, and instead demonstrate to us exactly how to do it right?”

        Willis, there are hundreds upon hundreds of papers that show how to do it. We aren’t in charge of your literature search. We aren’t. No, seriously, we aren’t. I’ve reluctantly given dozens of hints. Reluctant because I’m concerned that you might figure something out and confuse people in new ways.

        Thanks, Dino. In other words, you will NOT demonstrate to us how to do it right, so you’re of no use to anyone.
        Regards,
        w.

    • Reply to willis ==> Well, at least you almost asked a civil question….
      First, let’s look at the question of your “definition” of a Chaotic System.
      Here are the references you give, expanded to get a better idea of the context:

    • [well, something went wonky on the last try…one more time…]
      Reply to willis ==> Well, at least you almost asked a civil question….
      First, thank you for supplying references — now, let’s look at the question of your “definition” of a Chaotic System.
      Here are the references you give, expanded to get a better idea of the context:

      definition
      and
      L_E

      In the both cases, it is clear that what diverges are two orbits with nearly identical initial conditions in the phase space of strange attractors. And, of course, both point to the central point of sensitivity to initial conditions.
      In none of the data-sets proffered are “today’s value” the initial condition for “tomorrow’s value” — for example, temperature today at the Royal Observatory does not cause the temperature tomorrow at the same place; in the sense that today’s average temperature is not even an important input into the formula that determines tomorrow’s temperature …. while the average temperatures of a nearby air masses, the local atmospheric flows and the local values for cloud dynamics are causative inputs.
      In the same way, today’s Nile River flow value is not an initial condition for next weeks or next months flow…they are both results of some complex dynamical system — and flow of a major river is a naturally, physically “averaged” metric in the sense that it is the combined product of many local flows of tributary streams and rivers each depending on very local conditions.
      None of the data-sets represent values of orbits in the phase space of strange attractors.
      I quote rgbatduke one more time:

      “It is also difficult to map out a strange attractor with just one variable and time — the “chaos” is probably evident only on a torus in some (probably large) dimensionality so that when you look at global average temperature in particular, you are seeing only a projective view of the torus into a single dimension, nothing much to see and (for a bounded torus) no divergences per se.
      So you are looking at the wrong thing, in the wrong dimensionality. “

      Now, this is not to say that “local daily high temperature at Armagh” is not chaotic — it may well be, but it is a very tightly regulated output. Lorenz points out that the local high temperature in Phoenix, AZ, in June, is just not going to be -10° C no matter what slight adjustments are made to initial conditions — the dynamic system that produces temperatures in Phoenix doesn’t allow it even if it is Chaotic — the Chaos may well appear in some other dimensional view of the dynamic system that produces Daily High Temperature of Phoenix, AZ.
      Your question:

      Are you willing to stop endlessly claiming I”m wrong, and instead demonstrate to us exactly how to do it right?

      My answer:
      Part 1 (the “Do you still beat your wife?” portion) — I am not claiming, I am pointing out the same things that Robert Brown points out — your approach and your chosen data-sets are inappropriate for the purpose of answering the question “Is the Earth’s Climate System chaotic?”.
      Part 2 — Note: Here you have violated the ‘deal’ : “(not a demand that I solve your conundrums or misunderstandings about Chaos Theory)” — Short answer ==> Not my job. [Free hint: see rgb’s quoted comments.]

      • Thanks, Kip. Here was my simple request, which you have refused:

        Are you willing to stop endlessly claiming I”m wrong, and instead demonstrate to us exactly how to do it right?
        That is to say, if you think I’m using the lyap_k function incorrectly, then show us how to use it correctly. Or if you think I’m using the wrong datasets and methods to see if the climate is chaotic, then point out the correct datasets and the right methods and post your data and code.

        In other words, you are NOT willing to demonstrate how to do it right, and thus you, like Dinostratus, are of no use to anyone.
        Seriously, guys. Endlessly telling someone that they are wrong means nothing unless you can show us that you know what you are talking about by demonstrating exactly how to do it right. The fact that you both refuse to do so speaks volumes.
        Sorry, but I’ve got better things to do than be abused by people who are unwilling to show us the right way to do things.
        w.

      • Reply to Willis ==> I promised to answer any single civil question that you chose to ask — not accept whatever dare or demand you chose to throw at me.
        I did so, the answer was “Not my job” (I could have used more words and said “It is not my job to demonstrate to you exactly how to do it right and I have no intention of doing so.”)
        It is your project — just do it and let us know the results.
        RGB tried to steer you in what he feels is the right direction, and gave a suggested course of immediate study, for both himself and for you, in pursuit of a solution to the question of “how to do it right”. He is a lot mathy-er and smarter than I am, so my advice would be to follow his advice.
        Looking forward to your next essay on the topic.

  64. Willis has come to the table with mathematics and calculated results. The mathematics are the standard methodology for investigating the Maximal Lyapunov Exponent (MLE). Because of the exponential growth of initially nearby states the response is determined solely by the MLE as the effects of the other components will be annihilated.
    This Google Scholar search leads to thousands of hits related to calculation of the MLE for both simple lists of data and ODE systems. In order for Kip and Dinostratus to invalidate the methodology used by Willis they are required to show that his methodology is not among these methods.
    Kip and Dinostratus have arrived at the table with nothing but hand and arm waving. Neither of them can supply a single citation in which the chaotic response is characterized solely by the fractional dimension of the trajectories of the response.
    The literature, on the other hand, is filled with peer-reviewed reports and papers in which methodologies to calculate the MLE, and applications of the methodology to various simple non-linear ODE systems.
    More important by far is the fact that neither of them has suggested what the fractional dimension of the trajectories from CGM calculations might be. And most important of all, neither has suggested how that dimension might be calculated.
    In fact, they have yet to cite a single paper in which the fractional dimension of the original 1963 Lorenz equations has been calculated. I’ll supply one for them, and the paper indicates that this computational task is far from trivial. There’s a little difference between calculation of the trajectories and calculation of this property. 🙂
    Climate Science is filled with references to the chaotic response of Earth’s climate systems. Yet the only indications that Earth’s climate systems represent chaotic response are the numerical results of GCM calculations. These sole indications, in others words, can at the most only provide the characterization that Earth’s climate systems are Numerically Chaotic.
    It is critically important that results of GCMs are demonstrated to be a high fidelity representation of the physical domain that they are designed to simulate; it’s called Validation. Demonstration that the Numerical Chaos of the GCM calculations is present in the physical domain is critically important. One reason that this demonstration is important is the well-known fact that numerical solution methods have a potential to introduce chaotic behavior into equation systems that cannot exhibit such response. The numerical approximation to a single linear equation, for example. This unusual, and absolutely wrong, behavior even includes the bounded response that is an important aspect of chaotic behavior. Typically, numerical instability shows divergence and blow-up of the calculations. Some of the reports and papers listed here include demonstration of incorrect chaotic response due solely to numerical solution methods.
    Likewise, accurate numerical resolution of even the simple original 1963 ODE system of Lorenz is far from trivial. Check the literature citations in that paper for additional information. The paper was ultimately published in Tellus, IICR.
    Other introductory material related to numerical solutions of ODE systems that exhibit chaotic response is available here. Most of that information dates from 2007, 2008. There are very likely more recent resources available if you’ll Google around.
    Mathematics and results on the table trump hand and arm waving every time.

    • Reply to Dan Hughes ==> Correctly calculated mathematics applied to the wrong sort of thing does not produce a result that even needs to be refuted.
      You must have missed my very first comment on this essay:

      “This conclusion:

      “however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest.”

      is true.
      “It is also trivial if the question is “Is Earth’s climate system “chaotic”? [in the strict sense of Chaos Theory/Non-linear Dynamical Systems].”

      You seem to be “mathy” — read Dr. Robert Brown’s several comments above (he comments here as rgbatduke) who raises the self-same issues that Dinostratus and I raise, in his own inimitable, more-mathy-than-I, style. (If you use the Firefox browser, you can Cntrl-F opening a small search box at the bottom of the page and type in “rgbatduke” to find his comments.).
      There has been no invalidating of the mathematics involved — I’m quite sure that the r-function for calculating MLEs does so correctly — it is this whole approach to this specific selection of data-sets that is wrong if the question being asked is, in fact, as Willis claims “Is The Climate Chaotic?”
      As Dr. Brown states more succinctly than I:

      “So you are looking at the wrong thing, in the wrong dimensionality.”

      If, after reading all of rgbatduke’s comments, you still have questions, I’ll try to answer them.

      • None of the citations in this comment are concerned with experimental data. Instead several general aspects of chaos and numerical methods were investigated in the reports at the links.
        Apparently you again missed the critical step of actually reading the information before commenting. Had you bothered actually reading the information you would have some points that are in agreement with Brown’s initial investigations.
        I’ll await your appraisal of the applications and results of others as summarized in my comments below. You might find the information at the links useful.

      • Reply to Dan ==> Of course the real, in-depth stuff agrees with a lot of Dr. Brown’s points — he is often right.
        Please note, in case you haven’t noticed yet, that I do not respond to demands that I do something particular with the comments of others. I answer questions if I find them civil, interesting and I think I have something to say that would add to the general conversation. If you ask such a question, then I may or may not answer it, just as we all do in everyday conversation. [The only difference here is that we type instead of speak and there are a lot of people eavesdropping.]
        Adding in insulting personal negative assumptions [re: intentions of others, what they have or have not done outside of your sight, etc] to comments directed at me gets the same response I give boors at parties and conferences — I ignore them and move on to others who know how to carry on conversations with strangers.
        If you want to start a new conversation about “Is the Climate Chaotic?”, write an essay and we’ll all discuss it in its comments section.

      • ok, I have a question about this statement by Dr. Brown:
        “From this I conclude that it is not at all implausible to assert that the fond hope of the GCM modellers that using a stepsize 30 orders of magnitude larger than the Kolmogorov scale will even qualitatively reproduce the distribution of possible future climate trajectories may be unreasonable, as I think has been pointed out before by several people but without quantitative support.”
        My question is: Which of the fundamental equations employed in GCMs require consideration of time and space scales on the order of the Kolmogorov scales, and what are the fundamental physical phenomena and processes critical to the applications objectives of GCMs that require such consideration?
        Although I was initially instructed to ask any specific questions about Dr. Brown’s posts, I have now been informed that it must be in the form of a just-so question. As just-so is not well defined, it is very likely that Kip will not respond.
        When a response to a post fails to address any aspects of the material cited in the post and provides a statement that is not in any way related to the material in the in the post, I consider it a most excellent assumption that the responder has failed to read the post.

      • Kip Hansen said:
        “Reply to Dan Hughes ==> Correctly calculated mathematics applied to the wrong sort of thing does not produce a result that even needs to be refuted.
        You must have missed my very first comment on this essay:”
        And then he said:
        “Adding in insulting personal negative assumptions [re: intentions of others, what they have or have not done outside of your sight, etc] to comments directed at me gets the same response I give boors at parties and conferences — I ignore them and move on to others who know how to carry on conversations with strangers.”
        Emphasis mine
        Apparently the second emphasis of Kip’s words above is a self reference. Thus he is a self-referenced boor. :: QED

      • Reply to Dan Hughes ==> I suggested that you read Dr. Brown’s comments so you wouldn’t have to go over the ground already covered by him. If you have questions about something specific that Dr. Brown has said, ask him. He also sometimes responds to interesting questions.
        I look forward to your essay on Climate and Chaos.

    • “In fact, they have yet to cite a single paper in which the fractional dimension of the original 1963 Lorenz equations has been calculated. ”
      I gave the wiki for a list of dozens of them.
      Try again. 🙂

      • In this Wiki article I count 16 References. The text cites a single one of these, [13] Grassberger and Procaccia, relative to the fractal dimension.
        One (1) is somewhat less than “dozens”, which implies multiples of 12. So is 16 somewhat less than “dozens”, for that matter.
        Not to mention that the comment referred to you and Kip, not to some implicit third-order removed citation, the details of which you were apparently not aware.

      • Nope.
        Your accusation was “In fact, they have yet to cite a single paper in which the fractional dimension of the original 1963 Lorenz equations has been calculated. ” Yet I gave a link to 16 fractional dimensions (thanks for counting them!). You were so sure you were right, you failed to understand what you were arguing against.
        Next time at least examine the possibility that the reason you disagree with someone is that you are wrong.

      • No! Wait! Not only did you not read what I wrote, you didn’t even link to the right wiki article I gave. Getting flustered? I hope so. It would show some self awareness.
        I said this, “The Lorenz attractor, for instance, has a Hausdorff dimension of 2.06, according to wiki https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension . It covers more than an area. It fills less than a volume. It has a fractional dimension, ergo it is chaotic.”
        You said this, “In fact, they have yet to cite a single paper in which the fractional dimension of the original 1963 Lorenz equations has been calculated.”
        You are wrong. Eat it.

  65. Here’s an interesting statement from Giovanni Giacomelli, Stephane Barland Massimo Giudici, and Antonio Politi (2010), “Characterizing the Response of Chaotic Systems”, Physical Review Letters, 104, 194101.
    The last sentence of the Abstract reads, “The exponent [gamma sub 1] is a dynamical invariant, which complements the standard characterization provide by the Lyapunov exponents.”
    I changed the Greek symbol for gamma with subscript 1 to the words gamma sub 1, and the bold is mine.
    hmmmm, standard characterization provide by the Lyapunov exponents

  66. Here’s an interesting report: H. Abarbanel, S. Koonin, H. Levine, G. MacDonald, and 0. Rothaus (1991), Issues in Predictability, The MITRE Corporation JASON Program Office Report JSR-90-320.
    ABSTRACT Since the beginning of the greenhouse debate, policy makers have demanded from the scientific community predictions of future climate in limited geographical areas and limited time intervals. Current climate models clearly do not have such capabilities, as is demonstrated by large disagreements among the models of continental size regions. Largely lost in the debate are fundamental questions such as: What is meant by predictability? What can be predicted and over what time and length scale? What errors can be expected from predictions? This report explores some of the issues by analyzing toy models of climate and existing data sets of global annual average surface air temperature.
    7 SUMMARY COMMENTS The question as to the origin of the low-frequency components in climate records – chaotic motion or predictable fluctuations – remains open. Statistical analyses of data sets and of low-order climate models strongly suggest that the low-frequency components are chaotic, but this hypothesis has not been tested on Global Circulation Models. Such tests are essential if progress is to be made with respect to the question: Is climate predictable?
    The theory of prediction for low-order dynamical systems is still in its infancy, but may have important applicability to the problems of short-term climate prediction. Developments in the field of nonlinear predictability should be closely watched since this is a rapidly growing area of nonlinear dynamics.
    We have emphasized the reduction of the attractor dimension by averaging in order to increase the prospects for prediction. The results obtained from long runs of GCMs should be tested along the lines described in this report in order to investigate the extent to which climate is predictable.
    Our report deals only with temporal averaging. Many of the questions raised by examining temporal averaging are relevant to the question of spatial averaging. The issues need to be investigated in three-dimensional models to better define the length scales over which prediction is possible.

  67. Here’s another interesting paper: I.M. Echi, E.V. Tikyaa, and B.C. Isikwue (2013), Dynamics of Daily Rainfall and Temperature in Makurdi, International Journal of Science and Research, Vol. 4, Issue 7, July 2015.
    Abstract: Having a detailed knowledge of rainfall and temperature dynamics is important for an adequate management of our meteorological and hydrological resources. Chaos theory being the basis for studying nonlinear dynamic systems has opened a lot of doors towards understanding complex systems in nature such as the weather. In this study the dynamics of daily rainfall and temperature in Makurdi from January 1st 1977 – December 31st 2010 is investigated using chaos theory. A variety of nonlinear techniques such as power spectrum, phase space reconstruction, Lyapunov exponents and correlation dimension are applied. The optimal delay times were calculated for rainfall and temperature using the average mutual information technique while optimal embedding dimensions were also obtained using the method of false nearest neighbors. Phase space reconstruction was carried out with these optimal values using the method of delays. The phase portraits showed geometry of distinct shapes interwoven to form spongy like structures indicating the presence of randomness and chaos in the data with that of rainfall concentrated at the origin due to the numerous zeros in the rainfall values. The correlation dimensions were estimated using the Grassberger-Procaccia algorithm and found to be 1.02 and 5.82 for the rainfall and temperature respectively while the Lyapunov exponents were calculated using Rosenstein’s approach and obtained as 0.00832/day and 0.00574/day for daily rainfall and temperature respectively. The small values (υ<20) of the correlation dimensions obtained suggest the presence of chaotic behavior of low dimensions; inferring that the rainfall dynamics is dominantly governed by a minimum (maximum) of 2(18) variables while the temperature dynamics is governed by a minimum (maximum) of 6(17) variables. The positive values of the largest Lyapunov exponents confirm the presence of chaotic dynamics in rainfall and temperature records over Makurdi and suggest predictability of within 112 days and 174 days for rainfall and temperature respectively.
    Keywords: Chaos, power spectrum, phase space reconstruction, phase portrait, correlation dimension, Lyapunov exponent.

  68. The frequently misunderstood Lyapunov exponent is intrinsically a measure of stability of dynamical systems, rather than a patent property of arbitrarily chosen experimental data or time series. For reliable determination of LEs, more than one phase space trajectory of the solution of the governing DEs is required. Although various heuristic algorithms have been developed for estimating the maximum LE from a single data series (e.g. Wolff), various experts in chaotic system behavior (see, inter alia, Sprott, as well as interhttp://chaosbook.org/chapters/Lyapunov.pdf ) caution strongly against blind numerical calculation, while citing the Perron effect as an obstacle to reliable detection of chaos in any case.
    Nevertheless, the existence of chaotic behavior within the climate system is amply evident in the turbulent fluid motions of oceans and the atmosphere. It is the highly non-linear Navier-Stokes equations, rather than the equations of classical thermodynamics, that manifest a chaotic regime ubiquitously encountered on the spatio-temporal scales of weather. But when great spatio-temporal averaging is applied to the basic field data, it’s a small wonder that a search for indications of chaos in various “climatic” series turns into a fool’s errand.

    • The last sentence in the Abstract of the paper cited just above your comment says:
      “The positive values of the largest Lyapunov exponents confirm the presence of chaotic dynamics in rainfall and temperature records over Makurdi and suggest predictability of within 112 days and 174 days for rainfall and temperature respectively.”
      It seems to me that this example is not consistent with the last sentence in your post.
      Are there deficiencies in that paper?
      The example also seems to be inconsistent with several comments above in this thread regarding use of time series of a single quantity of weather/climate data.

    • Nice reply.
      I do take issue with “It is the highly non-linear Navier-Stokes equations, rather than the equations of classical thermodynamics” as the highly non-linear Navier-Stokes equations fall under the umbrella of classical thermodynamics.

      • You will be hard pressed to find a classic thermodynamics text that features Navier-Stokes. They are a staple, however, in fluid dynamics.

      • I just looked in wiki for a good definition of “clasical thermo.” Apparently the line is drawn more conservatively than I thought. I had the line between fluid type thermo and quantum physics type thermo. The pendants create a third class and what I considered classical thermo does not apparently include quasi or near quasi steady state thermo. That sits outside a strict definition of classical thermo. I’ll have to remember that the next time I consider acoustical impedances. That was classical to me but apparently it’s borderline depending on the time constants of the process.
        See? Even I can learn something.

  69. This paper presented the Rosenstein, Collins, De Luca method for calculating the MLE. The abstract states:
    Abstract
    Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

    The Introduction also contains the following:
    Dimension gives an estimate of the system complexity; entropy and characteristic exponents give an estimate of the level of chaos in the dynamical system.
    The Grassberger-Procaccia algorithm (GPA) [20] appears to be the most popular method used to quantify chaos. This is probably due to the simplicity of the algorithm [16] and the fact that the same intermediate calculations are used to estimate both dimension and entropy. However, the GPA is sensitive to variations in its parameters, e.g., number of data points [28], embedding dimension [28], reconstruction delay [3], and it is usually unreliable except for long, noise-free time series. Hence, the practical significance of the GPA is questionable, and the Lyapunov exponents may provide a more useful characterization of chaotic systems.
    For time series produced by dynamical systems, the presence of a positive characteristic exponent indicates chaos.

    I do not see in this paper, or in any of the few papers that I have looked into that results from this Google Scholar search any indications that the methods cannot be applied to experimental single time series of experimental data.
    De Luca has built software for estimating Lyapunov exponents for systems of ODEs.

    • Systems of ODEs are a far cry from mere experimental time series of a single state variable. That crucial distinction seems to have been lost in your view.

      • The paper uses both single-variable algebraic chaos and simple chaotic ODE systems. The methodology is not restricted to only one or the other.
        Apparently, actually reading the cited paper is lost on you.

      • The paper uses both single-variable algebraic chaos and simple chaotic ODE systems. The methodology is not restricted to only one or the other. The results of De Luca’s work on ODE systems are reported in other peer-reviewed papers. The software is available online.
        Apparently, actually reading the cited paper is lost on you.

  70. Yes, and so we have finally reached a determination that averaging over some temporal span is ok.
    If it rains continuously every pico second for 24 hours every day for some number of days, how is the allowable averaging period determined?
    So far as I know there are no theoretical guidelines for limitations on temporal averaging.
    So far as I know there are no theoretical guidelines for limitations on spatial averaging.
    And so far on this thread, no one has offered any theoretical guidelines for limitations on averaging over either time or space.
    Only hand-waving ad hoc like statements have been offered.
    Can anyone point me to a paper that offers theoretical investigations of the limitations of applications of the method?
    This this Google Scholar search might be a good place to start.

    • Inasmuch as averaging obliterates spatio-temporal variations on scales smaller than the averaging interval, the suppression of the effects of chaotic fluid motion in grand averages should be self-evident.
      Dynamical systems are the subject of intense ongoing research, conducted by highly qualified analysts. It’s unreasonable to expect google searches or blog comments to provide ready-made theoretical guidelines to satisfy the curiosity of nonspecialists or analytic amateurs. There are good reasons, however, why Sprott (see http://sprott.physics.wisc.edu/chaos/lyapexp.htm) opines that “When one has access to the equations generating the chaos, [finding the GLE} is relatively easy to do. When one only has access to an experimental data record, such a calculation is difficult to impossible…”
      Blindly turning the computational crank via ad hoc algorithms can produce numerical results; their significance, however, always remains highly questionable.

  71. Re: 1sky1 10/28/15, 12:41 pm:
    Blindly turning the computational crank via ad hoc algorithms can produce numerical results; their significance, however, always remains highly questionable.
    Well said. Another example is the set of estimates and calculations of the Equilibrium Climate Sensitivity, defined as the rise in temperature following a doubling of atmospheric CO2 concentration. What is measured is the rise in temperature and what is estimated is the doubling of ACO2, all right, two numbers available at all times. What is missing, though, is that it is not temperature following CO2, but the reverse.
    The discussions here about the fine points of chaos remind me of the elaborate and detailed discussions of, for example, ocean carbonate chemistry, aerosols, and radiative transfer. Each group of specialists strives to perfect his local region of a vast mobile, no one looking up to notice that the model is not connected to the ceiling.
    For chaos, however, the analysis immediately runs into its own peculiar disconnect. The climate, being a subspace of the natural world, is not deterministic, besides having no equations, no initial conditions, not even a coordinate system. Those are properties of climate models. By definition, chaos is a property of models, not of the real world. But keep turning the computational crank, guys.

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