Chaos & Climate – Part 2: Chaos = Stability

Guest Essay by Kip Hansen

 

 

intro_bifurct“…we should recognise that we are dealing with a coupled non-linear chaotic system, and therefore that the long-term prediction of future climate states is not possible.”

– IPCC TAR Chap 14, Exec Summary

 

 

Introduction:

The IPCC has long recognized that the Earth’s climate system is a coupled non-linear chaotic system.   Unfortunately, few of those dealing in climate science – professional and citizen scientists alike – seem to grasp the full implications of this. It is not an easy topic – not a topic on which one can read a quick primer and then dive into real world applications.   This essay is the second in a short series of essays to clarify the possible relationships between Climate and Chaos. This is not a highly technical discussion, but a basic introduction to the subject to shed some light on just what the IPCC might mean when it says “we are dealing with a coupled non-linear chaotic system” and how that could change our understanding of the climate and climate science.  The first part of this series is here: Chaos and Climate – Part 1: Linearity. Today’s essay covers a single common feature of non-linear chaotic systems: Stability.

 

Definition:

 

For the purpose of this essay, I use the following definition of a non-linear system:

Non-linear system: A system in which alterations of an initial state need not produce proportional alterations in any subsequent states; one that is not linear.

This broadly general, non-technical, definition is taken from The Essence of Chaos written by Edward Lorenz, the father of chaos theory. From Lorenz’s introduction to that book: “Some thirty years previously [1960], while conducting an extensive experiment in the theory of weather forecasting, I had come across a phenomena that later came to be called ‘chaos’ – seemingly random and unpredictable behavior that nevertheless proceeds according to precise and often easily expressed rules”.

I note for those with a more technical bent that there are many other definitions of “non-linear” and “non-linear systems” – in mathematics, physics, and engineering.   But, as I said, this is not a technical discussion, but a practical discussion.   One will find that for application to the field of weather and climate, Lorenz’s definition is perfectly suitable. Although Lorenz made the following statement about a loose definition of “chaos”, it might be applied to his definition of “non-linear system” as well: “My somewhat colloquial definition may capture the essence of chaos [non-linear systems], but it would cause many mathematicians to shudder” [one might add: philosophers, statisticians and engineers].

Let’s note a few things about the definition before we get too far:

The first is that it speaks of altering an initial state, the state from which the system begins. In a dynamic system, the initial state can be “at the start of our experiment” or, more often, it is the state of a dynamic system at any given instant as the “initial state” for either the next instant or any subsequent length of time the observer or experimenter might want to consider; today’s weather as the initial state for the prediction of tomorrow’s weather, today’s weather for the calculation of the expected climate a hundred years from now, today’s market-close prices on the stock exchange for the starting prices tomorrow and/or the state of the economy next year,    exactly where one starts the ball rolling down a hill and how hard one pushes it to get it started. All these can be seen as initial states.

Lorenz says “alterations of an initial state need not produce proportional alterations in any subsequent state”. It is important to note that he does not say that alterations do not, he does not say alterations can not, but rather, need not “produce proportional alterations in any subsequent state”.   (Proportionality sets non-linear systems apart from linear systems in which alterations in the initial state necessarily produce proportional alterations in subsequent states.) This allows that small changes in initial states may produce small changes in subsequent states, they can, but they don’t have to. In other words, they need not….they might, they may, they could, and in fact often do, produce large changes in subsequent states.

How this simple definition plays out in the real world is the subject that came to be erroneously called Chaos Theory. Erroneously because it is not a theory at all, but a broad field of study that evolved, in a practical sense, from Lorenz’s discovery and his subsequent paper Deterministic Nonperiodic Flow, published in 1963 in the Journal of Atmospheric Sciences. [While the majority of this paper is highly technical and requires advanced maths skills, the link is to a .pdf and I highly recommend reading at least the first two paragraphs of the Conclusion section of this paper – which is easily understood by laymen such as I.]

In the following sections, I will lay out just one of the common features of the behaviors found in non-linear, dynamic systems. The fact that there are common features may seem counter-intuitive – one may object to the idea that non-linear systems that produce chaotic, unpredictable results could have common features. The fact that they do is what attracted my attention to ‘Chaos Theory’ in the first place. For those of you who have doubts – please take the time to find out for yourself by reading Dr. Lorenz’s book linked above, or either James Gleick’s CHAOS – Making a New Science or Ian Stewart’s Does God Play Dice?. Or at a minimum, take a quick read through the Wiki page on chaos theory which will give you enough information to benefit from this essay.

 

 

Stability — A Common Feature of Non-linear Chaotic Systems

 

Listing stability as the first of these common features might come as a surprise to many who are familiar with non-linear systems and chaos – but not to engineers. As we were told in the comments of Part 1 of this series, engineers use designs, out of necessity, which are based on non-linear equations all the time and find them perfectly stable. One of the more savvy engineers pointed out that engineers, as a class, are naturally well aware of the instabilities – the turbulences – that can develop in non-linear systems, and so use values for those systems that are well below the points which tend to devolve into turbulence or instability. Thus knowingly or not, they are taking advantage of this property of chaotic, non-linear systems: Stability.

In a comment to Part 1 of this series, Leo Smith wrote:

“In fact, as practicing engineers we go to enormous lengths to keep our systems linear, or if we can’t arrange that, at least keep them out of chaotic regions. One designs a car to get to journey’s end, not fly into a thousand pieces at the first corner.

…. the general case of a non linear dynamic system is that if it does exist in nature, its probably chaotic. Linear and non-chaotic non-linear systems are very much the minority in Nature, but of course, because engineers seek predictability, they are the rule in most engineering.”

This image, similar to the one used as the introductory image today, helps us to understand this point. I’m afraid you’ll have to trust me for the time being that this graph is common to many [all?] non-linear, chaotic systems in the real world – I admit that it doesn’t seem reasonable – but a great deal of research has been done on it over the last 50 years, and it is so. Versions of this diagram appear in all of the studied real world dynamical systems that I am aware of … I give just one example further on.

bifurct

Figure 1: “Bifurcation diagram of the logistic map xr x (1 – x). Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.”

 

The line that extends to the left of the graph continues as a single line, sloping in a curve downward as r approaches 1 (see the graphic at the beginning of the essay).

When we graph this equation — xr x (1 – x) — with a beginning “r” of 2.8, and an initial state value of 0.2, this is what we find:

per1_8x2

Even though the starting value for x is 0.2, iterating the system causes the value to x to settle down to a value between 0.6 and 0.7 – more precisely 0.64285 — after 50 or so iterations. Jumping in at the 50th iteration, and forcing the value out of line, down to 0.077 (below) causes a brief disturbance, but the value of x returns to precisely 0.64285 in a short time:newplot3

Kicking the value out of line upward at year 100 has a similar result. Adjusting the “r”, the forcing value, down a bit at year 150 brings the stable attractor lower, yet the behavior remains stable, as always.

Refer to Figure 1, the bifurcation diagram, to see that the value 2.8 for r corresponds with a value of x of 0.64285. Had we used 2.4 for r, the stable value of x would have been 0.58333.   All of the values from 1 approaching 3 produce a stable single value result.

This single value result is an “attractor” for the system. An easy way to think of a single attractor is to think of a funnel.  Liquid poured anywhere inside the rim will be drawn to the bottom (by gravity) and exit in one, and only one location, the spout.

 

funnel_attractor

 

Alternately, consider the bowl. A marble placed anywhere inside the rim and released will roll to the bottom – it may roll around a bit, up the other side and then down again, but eventually comes to rest at the bottom of the bowl – that space, the bottom of the bowl, is the single attractor for that dynamic system.

nut_bowl

 

When I was a kid, we had a nut bowl similar to the one pictured above, except that ours had a common rim that was 2 inches higher than the interior separators. We spent hours rolling marbles down the side from different points, like the ball in a roulette wheel, betting on which depression the marble would end up in. This nut bowl, with four separate bottoms but a common rim could be said to have four marble attractors. A roulette wheel could be said to have 37 attractors, 0 through 36 (or 38 in Las Vegas, where there is an additional 00 position).

We will take up the subject of attractors in a later part of this series but we should understand that the characteristic of stability in non-linear dynamics stems from the function of attractors. The bifurcation diagram gives the attractor for each value or “r” from 2.4 to 4.0. Some regions have a single value attractor, many are periodic with more than one – 2, 4, 8, 16, 32 and 3, 6, 12, 24 — stable attractors , and some regions are chaotic (we will have more on chaotic regions in a future installment of this series).

Where do we see such dynamic systems? The formula used here is the same as the basic formula for Population Dynamics, where x is the unit of the carrying capacity of an environment (in which 1 would represent 100%) and r is the growth factor. With low growth factors, populations tend to be stable and less than the carrying capacity of their immediate environment. This illustration for the fictional May Island Squirrel Population was included in Part 1. [ The Predator-Prey Equations, which reflect some populations, are also non-linear and are usually linearized in order to find approximate solutions.]

may_island_pop_final

We see in this diagram the population of squirrels on May Island. With a growth rate below 3 – 2.7 in this example (green  blue trace) – stabilizes at 0.6xxx, roughly 60% of the islands carrying capacity. Bumping the growth rate up to 3 (orange trace) causes the population to wobble, saw-toothing a little bit higher. The wobble increases as the growth rate increases until at a growth rate of 4 (green trace) the population is very unstable, chaotic, and suffers a possible extinction (or very low numbers) for years 14-18 or so.   The purple line, provided for interest, represents what the average person would expect – a linear progression from a low population to a full carrying capacity population produced by a [incorrect] linear understanding of population dynamics.

In real world, complicated non-linear dynamic systems, such as the interplay between adult flour beetles, their pupae, and their larvae in a confined volume of flour – in which the adults happily cannibalize the pupae – we see this more complex version of [a small portion of] the bifurcation diagram which was then experimentally found to be valid.

Fig2_600

The researchers focused on the chaotic regions except at the value “1”.   We can see where we could expect stable single value attractor results in the region just above 0.1 and at all values above 0.45, probably some period of 3 (3, 6, 9…).

The characteristic we see in these examples is stability – even the period-two wobbles are stable around a central value.

In real life population studies, all these features are seen. Some species populations are more prone to developing growth rates that cause chaotic dynamics – such as deer population in rural NY State where I raised my children. With deer, two or three good years in a row result in a growth rate that spikes population to unsustainable levels, resulting in heart-breaking population crashes from winter starvation. As a result, the NY State Department of Environmental Conservation struggles to keep growth rates in check regionally through the issuance of additional deer hunting tags for females – chopping back growth rate directly.

Engineers can and do take advantage of the stability in non-linear systems by designing systems that fall within the stable portion of bifurcation diagram for the particular system – staying well away from the point at which wobble, turbulence, and chaos ensue. Not only are these non-linear systems stable – they embody stability as part of the dynamics – perturb them and they return “of their own accord” to their stable point.

In real world dynamic systems, there is always some element of randomness added into the system – breezes, vibrations, confounding elements of other systems, traffic in the street – always something that deflects the system from our perfect mathematical rendering above.   The characteristic of a stable attractor dampens these effects – as long as the system stays away from the point of bifurcation, that period doubling which leads to Chaos.

Here’s what a non-linear chaotic system running in stable mode looks like in the real world: stable_chaos_temp_numbered

The graph above shows a ~ 5 % or so variation over the entire length — it has jiggles and an apparent trend.   If this were a squirrel population somewhere, we’d suspect that things had generally improved a bit, perhaps increasing carrying capacity of the environment, a decline in predation, an increase in maturity of food providing trees or an incursion of a new food supply – say urban squirrel feeders. The point is that despite the multitude of things that could be affecting the overall system, it exhibits a high degree of stability, staying in a certain range.

It is a the same as the earlier blue graphs – Robert May’s population dynamics formula, a non-linear dynamic formula known to produce chaotic results when forced above a certain level (as in the green trace in the May Island Squirrel Population graph).   This version, though, was created by slightly altering the forcing (the “r” in the formula) so that it increases and decreases minutely — by 1/1000th a year, up or down in various time periods – and the entire result being overlaid with a random addition varying from 2 to 6 percent.

Anyone recognize the graph?  Ever seen this:

 Temp_1880_2014

A rough, and widely used, version of land surface air temperatures over the last 140 years from the Met Office.   Here’s what they look like overlaid:

Temp_overlaid

Of course, this is a set-up. This proves nothing, nor is it intended to prove anything. The purpose is just to illustrate that in the stable zone of a chaotic system, with very slight changes to the “forcing” factors and a bit of the natural jitters and randomness found in all real world systems, it is easy to make a generalized non-linear chaotic system behave in the same manner as the part of the Earth’s climate system that produced surface air temperatures over the last 145 years.

 

Chaos, Stability and Climate:

 

“The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.” – IPCC TAR Chap 14, Exec Summary

This much is not in doubt – “the [Earth’s] climate system is a coupled non-linear chaotic system” — nearly all sides in the Climate Wars recognize this simple-appearing fact. That is where agreement apparently ends.

Many have put forward the idea that the Earth’s Climate System is in a stable period-two state (see the bifurcation diagram above) – shifting on a very long time scale from Ice Ages to Interglacial Periods and back again. From this, we can get general agreement that the Earth is currently in Interglacial mode. Within each extreme, the climate appears surprisingly stable – nice solid well-constrained Ice Ages and nice solid well-constrained Interglacials.   The sun seems to slightly perturb the system in each state, with some Interglacials or portions of them being a few degrees warmer or cooler and the same within Ice Ages.

In more recent times, kicking the climate system with a decade of frequent volcanism causes things to chill right down but after a few years, the system is back into its normal range. The Roman and Medieval Warm periods cool off and after a bit of a “Little Ice Age”, things warm back up. This warm period, cool period, warm period may be the result of a bit of wobble developing in the system or due to slight changes in the strength of the Sun, orbital changes, and other natural physical confounders – generally referred to as “natural variation”.    It might be a sign that the system is “striving” for stability, “trying” to return itself to a single stable point.   It is possible that the climate for the least ten thousand years is the stable value overlaid with a bit of physical noise.

All_palaeotemp

source

The graph above (whether we accept it as accurate or not and ignoring the silly, unscientific additions of “future” values in red for 2050 and 2100) shows that for the last ten thousand years proxies of surface temperatures show a stability to within a range of 2°C, very well constrained.   The left-hand panel shows the period-two behavior of Ice Ages and Interglacials. (Note: the time scale changes at the panel break. Earlier panels of the original image have been cropped out. There are questions as to whether the vertical alignment of the first, left-hand, panel is properly represented and/or supported by data.)

The incredible thing about the Earth’s climate is that it is so stable, over the last million years, given the huge geological and biological changes that have taken place, including the most recent post-glacial sea level rise, which resulted in a massive change to ocean volume, shape, and extent – the oceans being one of the two coupled non-linear dynamic systems that make up the whole of the climate system.  Even with the Ice Ages, the climate has been stable enough for life to grow, change, and prosper.

Do we really know anything about why the geological-time temperature graph looks as it does? I’d have to say “No, not really.”   At best we can describe it: Ice Ages and Interglacials and on a smaller scale, the most recent 10,000 years, warm periods and cooler periods; we may have some guesses as to causes of the changes, some guesses more likely to be correct than others.

Chaos Theory gives us another way to look at the question based on the fact that Earth Climate is a complicated complex non-linear dynamic system, made up of at least two closely-coupled non-linear dynamic systems: the oceans and the atmosphere.   How that coupling affects the total system is what Climate Science is intended to study. How coupled chaotic systems behave in general, to my knowledge, is simply not known except in the studying of Earth climate as an example. In my opinion, the uncanny stability of the Earth’s surface temperature over the past 10,000 years is correctly represented as a single-value stable attractor of the coupled system.

 

Take Home Messages:

 

In this essay, I have tried to present the less-talked-about aspect of chaotic non-linear systems: their stability in certain regions, with a single-value attractor to which the system returns after being perturbed.

  1. Non-linear chaotic systems have regions that are not only stable but are high resistant to being de-stabilized – they return to their stable value, their single-value attractor, after being perturbed.
  2. The Earth’s land surface temperature record is extremely stable over the past 10,000 years. This stability may represent a single-value stable attractor for the system.
  3. The Earth’s Climate System is “a coupled non-linear chaotic system” – and while exactly what that coupling means one can or should expect from it is not yet well understood and remains unclear, we should look to Chaos Theory to provide the necessary insight we need to reach a better understanding.

 

# # # # #

 

Author’s Comment Policy:

I love to discuss this topic and I will try to answer questions to clarify any points I have made above.  Before asking more in-depth questions about Chaos Theory, please make sure you have read Part 1 of this series (I recommend reading through the comments as well – particularly my answers to questions and all comments by rbgatduke.) If you are less-than-up-to-speed on Chaos Theory in general, and want to know more, I recommend a quick read of the Wiki entry on Chaos and for those with deeper levels of interest, I gave an Intro to Chaos Theory Reading List.

I will not be defending the idea that the climate system is a coupled non-linear chaotic system – it is simply too well established and supported by the physical actuality. Those who wish to rail against this fact can have their say here, but I will not be replying. It would be interesting to read a well presented essay from those who hold such an opinion as to why they do so and I’m sure Anthony would post such an essay.

I would be interested in your guesses as to why I say in the last paragraph before the Take Home Messages: “made up of at least two closely-coupled non-linear dynamic systems”.   Might there be more than two? If so, what might the additional systems be?

Further, this essay is not about the details of the Earth’s climate, CAGW, AGW, or related nonsense. I will not fight the Climate Wars in the comments – please reserve that for some other essay or take it over to the NY Times’ Dot Earth blog where a lively battle is nearly always taking place.

Thank you for reading here.

 # # # # #

[This essay was edited at 8:30 Eastern Time Nov 24 2015 to correct minor formatting  issues and at 9:30 correcting the name of a  trace color  (h/t goldminor) – kh]

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144 thoughts on “Chaos & Climate – Part 2: Chaos = Stability

  1. This is posting just at East Coast bedtime. I’ll be around first thing in the morning to field any questions.

    Thanks for reading.

    • (Mr. Dr. Ms?) Hansen;

      Fascinating article and great heuristics. Your right, our climate remarkably stable dividing its time between two major states, namely glacial and interglacial that are stable within a few degrees C. So there must be attractors (2?) that lead to the state changes (within these two state there are additional states that show their own stability like alternating colder/warmer periods). Could it be said that there have been tipping points where we transition suddenly (mostly) to the non-current state? Much has been made in some circles that the global average temperature (rising) may well be approaching a tipping point. Is this a reasonable thought and what kind of state might we tip to? Curious as to your thoughts.

      TPG

      • Reply to TPG ==> Actually, if I must have a title, it would be Captain. I hold a license from the US Coast Guard.

        Transition Points: Yes.

        Tipping Points: Probably not in the sense it is usually used. See a fuller reply on this issue here..

        As you have already pointed out, there is an apparent period-two oscillation in the paleo-record. In the last 20,000 years, we have come out of the Last Glacial Minimum (LGM on the graphic) and warmed up to the current 11,000 year stable Interglacial, well-constrained to within a 2 degree or so range.

        The climate might well change back to an Ice Age, but what would cause that, we simply don’t know. In Chaos Theory there are thresholds, when the system moves from a single-value attractor to a period-two, then period-four, period-eight….period doubling it is called, which lead rapidly to the chaotic region.

        I am not aware of anything in Chaos Theory that aligns with the idea of Tipping Points as defined in the Wiki article on the topic.

  2. How about at least 5 nonlinear coupled systems: Ocean, Atmosphere, Earth’s core, Sun and the Heliosphere.

  3. Good overview, thanks.
    Tricky question at the end. Here’s my list of coupled systems:
    Atmosphere, heliosphere, lithosphere, hydrosphere, biosphere, cosmosphere….
    The list can go on and on and include a carbosphere but I wouldn’t put it near the top.

  4. Non-linear system: A system in which alterations of an initial state need not produce proportional alterations in any subsequent states; one that is not linear.

    So a good example of this is how Earth responds to a doubling of CO2 in the air. Earth’s absolute temperature does not double, nor does it even go up linearly, but presumably logarithmically.

    • No, what needs to be emphasised about the definition is that it is a non-linear response of a state to alteration in the same state variable (or variable set). eg temperature responding to temperature. As in the example x → r x (1 – x).

      • Reply to Nick ==> There is the question as to whether or not temperature does respond to temperature.

        This can be seen as both a trivial and an extremely deep question. But it is important when looking at the temperature aspect of climate through the lens of Chaos Theory. If temperature is just a by-product of some other aspect of the dynamical system — then it will not be prone to chaos itself. If it is not, as you say, the “x” in the example x → r x (1 – x) — then we will not see chaotic behavior in the measurement, that metric, except as it acts a sort of proxy for whatever part of the larger dynamic system that produces it. We might see surface temps running up and down with the Ice Age/Interglacial cycle, but it very well may not be a “driver” — it may not be the “x” — it may just be a necessary by-product.

        We certainly don’t in a serious way think that “today’s temperature” causes “tomorrow’s temperature”. (Not that they are not related at all, one is just not the cause of the other.) In this way, I don’t think that we can expect temperature itself to be more than a tell-tale for climate.

        (This was not as easy to explain as I had hoped — maybe you’ll help me out here.)

    • “So a good example of this is how Earth responds to a doubling of CO2 in the air. Earth’s absolute temperature does not double, nor does it even go up linearly, but presumably logarithmically.”

      No, because a logarithmic increase which varies at a set rate according to a given input is still “linear” for these purposes.

      In your description if you turn the amplifier knob one click, the sound produced increases by (let’s just say) 3dB each time, i.e., the volume doubles with every notch the dial is advanced. In a truly non-linear system though, the amplifier might increase three dB with any given click of the knob…, or it might go up 0.5dB, or 5.0dB, or it might not move at all, or maybe it even drops by 3dB.

      The point is that in a non-linear chaotic coupled system, you cannot predict the outcome based upon any singular input.

      • Chaotic systems generally have feedbacks often both positive and negative; continue in the line of the amplifier, let’s add a speaker in room, a microphone and have people entering and leaving at apparent random. The microphone provides positive Feedback of the output of the speakers to the amplifier, and the people and their clothing absorbs sound providing negative feedback and the random entering and leave perturbs the system. Now can we predict when the speakers will start squealing as we increase the volume?

  5. I read both Gleick’s and Stewart’s books in the early 1990’s. Both were good reads, though the Feigenbaum number doesn’t seem to have attained the importance that Gleick thought it would.

    I do have a couple of memories of the books. For gleick’s book, it was seeing a lot of figures credited to “James P. Crutchfield” and wondering why the name sounded so familiar – then realizing he was my dentist’s son. For Stewart’s book, it was wondering if he was the SF author – then noticed the chapter on Lorenz was titled “The Courage of his Convections” and ll doubt was removed.

    A lot of “chaos” has roots in numerical analysis, there are some sets of “ill-conditioned” equations that are very sensitive to numerical errors, which can be thought of as more or less a source of noise. In the real world we have all sorts of perturbations.

    My final word on this subject is: turbulence.

  6. Good article. Thanks for writing it. As we know, physical systems can have all sorts of nasty little things going on, including positive and negative feedbacks. The best we seem able do is keep up our linear approximations, but the key is to know where the system goes into distortion, or out of linearity.

  7. Specific predictions inside any chaotic system are impossible, but long term predictions about how that system will average out are possible. See any casino for direct proof of this.
    So if a scientist has determined that CO2 has a specific thermodynamic property that can be mathematically quantified, then short term predictions are irrelevant. Whatever random or cyclical responses the rest of the climate has can at best delay the manifestation of the extra CO2 or at worst exaggerate them.
    If as a roulette player I switched from a table with one zero (French wheel) to one with two zeros (American Wheel) I could still have a great winning streak. This doesn’t change the fact that a person will lose their money twice as fast on the table with the two zeros wheel.
    So all the arguments about chaotic non linear systems are essentially irrelevant to the faithful.
    Of course direct negative feedback are a separate issue. If you have a quantifiable negative feedback value for clouds, for example, to offset the positive value for CO2, then this simply gives a new equation to apply to the thermodynamics, but says nothing about the variation and chaos that will occur either side of the new calculated mean.
    So the physics question always comes back to the thermodynamic value given to CO2. The easiest way to show this value to be based on faulty assumptions and poor science is to transfer it to a study of the planet Venus and to see how appallingly this value performs to describe temperatures there.

    • Correct. You cannot calculate how much a change in CO2 will change temps any more than you can predict the height of the tides from the force of gravity. You can only predict based on past history and cyclical behavior.

    • The roulette wheel is my favorite table game. The first time that I ever tried my hand at it, I was very successful after I was able to formulate a line of attack while playing. That has held true ever since, where at the very least, I am able to pass a lot of time for a small amount of money invested.

      • “I am able to pass a lot of time for a small amount of money invested.”
        Don’t you really mean:
        You are able to pass a small amount of money for a lot of time invested .

      • It could be looked at that way. Yet I found it a much better alternative than losing 100 dollars at a blackjack table in 10 or 20 minutes. At a roulette table I could sit and play 30 dollars while sipping their drinks for an hour or two. Besides that I have had some nice wins at the table.

      • Reply re: Roulette and Blackjack ==> I luckily discovered at a young age that I would play until I lost everything in my pockets, everytime I sat at a game of chance. I have stayed away from gambling since then — too dangerous for me.

    • “… long term predictions about how that system will average out are possible. See any casino for direct proof of this.”

      I don’t see why that applies to climate. In the case of coin tossing, dice rolling and roulette wheel spinning we know the “expected value” before we start. This does not apply to the climate – so all bets are off!

      • I agree that a roulette wheel results are random but not chaotic. Perhaps a better chaotic example would be one that followed a “Hurst” distribution.

    • Surely, a roulette wheel is a poor example for predictability in a chaotic system. It is chaotic in behavior, but statistically determinate in that over time, the outcomes will converge on an equal distribution of results around the wheel. The range of outcomes is constrained.

      In a nonlinear system, as the system is perturbed more aggressively, the period-doubling cascade begins and degenerates into a much larger set of available states, within the constraints. A chaotic system is not statistically determinate in any real sense.

      I recognize that I may not be using terms that are technically correct, but I trust that the sense of my comment is clear.

    • Reply re: Roulette ==>

      The game of roulette is not chaotic in nature — not in the Chaos Theory sense. It is simply random, entirely ruled by the laws of chance.

      The results of non-linear chaotic systems are NOT random — they are entirely deterministic, each state completely predicated on the precious state and produced often, as in the examples here, by simple [appearing] formulas. The results of non-linear chaotic systems are, however, unpredictable — once a non-linear chaotic system enters a chaotic region, one can not know, cannot predict even approximately, what the result will be at the 50th iteration — one has to go through all the preceding 49 steps to get there.

      Non-linear systems in their chaotic regions no not obey the laws of probability. They are not random.. They do not result in a standard normal distributions.

      This is why I asked readers to read Part 1 of this series or to read the Wiki Chaos Theory page.

      IMPORTANT NOTE TO READERS:

      All uses of the word “CHAOS” and “CHAOTIC” in this essay refer to the special sense of the word as used in the subject of Chaos Theory. It does not mean “random”. Quoting the Wiki:

      QUOTE: Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

      “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.”

      END QUOTE

      • Correct about the roulette wheel. Coupled non-linear systems are not necessarily unpredictable, much of the phase space lies in stable regions where prediction is possible (stable nodes and foci, and also stable limit cycles), only in certain regions are the solutions chaotic.

      • “Non-linear systems in their chaotic regions no not obey the laws of probability.”

        That is not generally true. I knew a fellow who did his master’s thesis applying the Chapman-Kolmogorov equation to determine the evolving conditional probability distribution of the outputs of various chaotic maps, including the logistic map above. The distribution is definitely not Normal or Gaussian, but it can sometimes be determined very neatly.

      • Kip,

        Perhaps poker is a better gambling analogy? . . in that a given card’s impact on the game can be radically different depending on the lay of the cards at the moment it is dealt?

      • Reply to Bartemis ==> I’d like a peek at that thesis. It is not that one can not find some idea of the probabilities in a chaotic system — but they will have no relationship to anything we would recognize. “Determining the evolving conditional probability distribution of the outputs of various chaotic maps”, often in multi-phase spaces, is not something you or I will be using to determine our next bet on the roulette wheel and probably not be using to make predictions as to future climate states. h

        From our bifurcation diagram, we can posit near certain probabilities for many values of “r” — and can be certain that some values will never appear at all.

        This is not the same as the kinds of probability that would allow predictions of future states and not the kind of thing that is normally referred to as the “laws of probability”.

        In a nutshell, there are a lot in interesting aspects of Chaos Theory, and real wild elements of astonishing and unbelievable complex beautiful order in those chaotic regions, but they are not what one would predict from the Laws of Probability.

      • Reply to JohnKnight ==> I only intended thew roulette wheel to represent a physical analogy of a thing with multiple attractors when in use. A simple bowl or funnel has one attractor, a roulette wheel has 37 or 38.

        Nothing more than that.

        It has been instructive to some here to explain that the roulette wheel game is not chaotic at all, only random.

      • Kip – IIRC, the logistic map with r = 4 converges to a U shaped “probability” density between 0 and 1, with mean 0.5 and pseudo-independent samples. You can verify it yourself running a few thousand variates and doing a histogram and estimating an autocorrelation. The fellow I knew actually derived a functional form for that limiting density, but I do not recall it. I’ve got the paper somewhere in my archives, but it would probably take hours if not days to find it – my archive is itself rather chaotic.

        It qualifies as probabilistic in the sense that you can specify a distribution of likely occurrences at the nth step of the iteration, starting from some initial distribution for the starting value (e.g., a uniform density between 0 and 1). If you knew a gaming table were relying on the logistic map with r = 4 to determine outcomes, you would be much better off betting near 0 and 1 than 0.5.

        One may take the initial probability density to be uniform between 0 and 1, and plot the updated density after each step of the map. If you choose r to be some value less than 4, the density converges to a multi-modal one, approaching delta-functions at the attractors. Once you’ve gotten within the event horizon, as it were, of one of the attractors, the conditional probability collapses to a single narrow mode approaching a delta function at that point as the number of steps taken increases.

        I believe I recall he computed the densities by segmenting the initial uniform distribution into partitions. Then, you can compute the conditional distributions for each partitition, and sum them all up using the Law of Total Probability.

        I think the point is not so much that chaotic systems cannot be predictable – sometimes they can be to some extent in a statistical sense – but that nobody is considering the climate from that point of view.

      • Kip,

        I wasn’t actually referring to your mention of roulette in the article, but the mention by a gambler concerning it.

        That gambling “system” seems essentially “linear” to me, in terms of each fall of the ball into an “attractor” being straightforward probabilistically, and resulting in a win or loss with regard to each bet, and the math involved being relatively simple . . but in poker a given card dealt can result in anything from a huge loss to a huge win, or anything in between depending on the circumstances it “falls into”, in terms of fit in the hand, what the other other players are looking at, and betting sequences during the play of hand . .

      • Reply to JohnKnight ==> Yes, I see. It is an interesting point — the complexity of the game of poker, in which the players share a common deck of 52 (54?) unique cards and hold five cards each, with combinations making up “value” of the hands (but not directly linearly — four deuces beats three tens — yes, very complicated and complex (I think).

        Thus the value of the next card to be dealt depends entirely on which player receives it, what he discards to replace with that card, and the combination of the four original cards kept.

        I don’t think that the overall game system is chaotic. I’d have to spend some more time thinking about it to be sure.

      • Kip,

        My actual conceptual framework for the analogy came from bridge, but I was kinda going with the gambler motif . . What you guys are describing mathematically seems, to this mathematically challenged soul, similar to what “our” minds were often dealing with when I was a serious bridge player. (What you call “chaos” is the fun part ; )

      • Just as a note: theoretically you can predict a non-linear chaotic system, even in a chaotic region, if you can simulate it with enough precision. The problem is that any real-world system is usually so complex that the required precision is less than the bit precision of the simulation. The best that a simulation can do is to tell you where the stable regions are.

        It should be pointed out that not only is the climate chaotic, but the inputs are not entirely know or foreseeable. I would particularly point to cosmic ray effects on cloud cover, which depend on the Solar System’s particular location in the galaxy.

      • Reply to Paul of Alexandria==> “theoretically you can predict a non-linear chaotic system, even in a chaotic region, if you can simulate it with enough precision.” There is a lot of research being done to see if this is strictly true. Some things can be known — phase space filled by the chaotic region, where we might expect stable regions, etc. What can not be predicted in the value at any particular time in the future….where gthe system will be at year 100 from today. This is not, of course, the same as saying nothing whatever can be predicted. Once a chaotic system gets into a strange attractor phase, you can predict with 100% certainty that it will remain somewhere on that attractor at all times….as long as the system is not changed — where it will be on the attractor at year 100 or 10000 can not be predicted.

        And, yes, you are right, we do not know all the inputs, nor necessarily the proportionality of those we are aware of. I was thinking today (on a long Thanksgiving drive) as to whether or not the combined gravitational forces involving the Sun, Moon, and Earth are non-linear and how much, if any, they affect climate. Do the Sun, Earth, and Moon make up a classic three-body system? I didn’t come to any conclusions….

  8. The May Island system you described is based on the Lotka-Volterra equations. The condition at r=2.7 is a stable focus, and a small perturbation from the stationary state will result in an oscillatory return to that point. At r=3 the stationary point becomes an unstable focus resulting in a limit cycle, the amplitude of which increases with r. If you start at r=2.7 and reach stable operation if you then increased r to 3 you would find that the system would start to oscillate.

    • Thanks for that clarification. From the head post one might have inferred the process reached and remained one of the two possible “attractor” values, the choice depending on initial conditions.

    • Reply to Phil and Joe => The multi-trace diagram and its explanation were meant to make this clear. Populations oscillate between the two values – they do not settled down to one. In May’s Population Formula, values of “r” at 3 and above are unstable, periodic or chaotic.

      The Lotka-Volterra equations are the “other” population dynamics formula — the Predator-Prey Equations,.

      You might enjoy reading this paper by Robert May: “Simple Mathematical Models with Very Complicated Dynamics”. in which he discusses the formula I use in this essay.

      • Thanks for the tip.

        Although I can’t heartily recommend the book, I will say that Stephen Wolfram’s “A New Kind of Science,” too, has quite a bit to say about the emergence of complex behavior from simple rules.

      • Your traces showed the approach from a distant point outside the limit cycle, as I pointed out it is also possible to approach the cycle from inside from the stable value if r changes to 3+.

        I am familiar with Bob’s work, he worked on ecological systems whereas I worked on chemical kinetic systems. In your earlier post you referred to Bob becoming Dean of Research at Princeton, he later moved to Oxford and inter alia became Chief Scientific Advisor to the HM Government, president of the Royal Society and became a member of the House of Lords.

        There is an actual island where such systems have been studied, Isle Royale in Michigan, Moose originally populated the island over one hundred years ago and population dynamics such as predicted by May’s model were observed. In 1958 wolves arrived on the island and the system changed to a prey-predator one (i.e. Lotka-Volterra, although strictly a 3-parameter system, forage, moose and wolves). It appears likely that the wolves will die out and the system will revert to the original one (disease reduced the wolves below a sustainable level).

    • Reply to spangled drongo ==> IF we had a way to know the true values of the paleo-climate surface temperature record to thew accuracy of single-digit degrees C (which I do not believe we do), then one might be able to calculate such a figure.

      Lacking proper data to that degree of accuracy? ==> “not really knowable”

  9. TILT… I can’t do this tonight. Kip. I will read this tomorrow and maybe understand some of it.

    • Reply to optcon ==> And what did you think of that essay? Did you find it scientifically convincing?

      Read through the comments there. rgbatduke and I, among others, had serious objections, not to the math work done, but certainly to the the choice of data analyzed (which we did not and do not think could have been expected to be found Chaotic in the first place) and to the conclusion which far-overstepped the boundary of scope of the work — testing a few random data-bases for signs of chaos does not even speak to the question of whether the Climate System is chaotic or not; and even if it did (which it does not), declaring that the climate was not chaotic would require ignoring all the real world data that says the Climate System is a coupled, non-linear chaotic dynamic system.

      This phenomena is a problem for Science in general. The media hype up little studies with news legs (meaning that they will attract eyes on the blogs and online newspapers, make good TV News sound bites). Headlines that generally misrepresent the actual findings or stretch them out into some big important finding when in fact they are insignificant and preliminary, create a sense of “new discovery” or a “discovery reversing what we all believe” when in fact no new discovery has taken place…just something maybe interesting found that now needs to be put to the test of real science. This is seen every single day in the fields of health and nutrition – “Coffee saves lives” — “Coffee causes cancer” . The vast majority of these studies prove to be of no value whatever when they are finally put to the gold standard of a double-blind medical trial.

      “So which to believe?” Use your critical thinking skills to evaluate the evidence, find more evidence on your own, study up, decide what seems more correct, keep an open mind until you have enough data to make a personal preliminary decision on the matter. In other words, do the same you do for everything. Certainly, do not just believe the latest thing you’ve read.

  10. The ocean tides are chaotic and predictable. However they are not predictable using that used in climate models. Instead the tides are predicted using astrology.

    • “Moon waves,” Otto Petterson called them.
      They introduce a cyclic perturbation into the climate system. Resonance.
      See:
      “The old-timers are right–winters aren’t what they were. And the reason may be gigantic tides deep under the sea that apparently change the climate of the whole earth.”-https://wattsupwiththat.com/2008/07/25/who-knew-rachel-carson-climate-change-expert/

  11. Kip Hansen:

    I write to add the congratulations in this thread for your eloquence, accuracy and clarity in your above essay.

    I have often pointed out the importance of what you have written above and – in hope of adding to thoughts generated by your essay – I again copy my explanation here.

    The climate models are based on assumptions that may not be correct. The basic assumption used in the models is that change to climate is driven by change to radiative forcing. And it is very important to recognise that this assumption has not been demonstrated to be correct. Indeed, it is quite possible that there is no force or process causing climate to vary. I explain this as follows.

    The climate system is seeking an equilibrium that it never achieves. The Earth obtains radiant energy from the Sun and radiates that energy back to space. The energy input to the system (from the Sun) may be constant (although some doubt that), but the rotation of the Earth and its orbit around the Sun ensure that the energy input/output is never in perfect equilbrium.

    The climate system is an intermediary in the process of returning (most of) the energy to space (some energy is radiated from the Earth’s surface back to space). And the Northern and Southern hemispheres have different coverage by oceans. Therefore, as the year progresses the modulation of the energy input/output of the system varies. The variation causes global average surface temperature anomaly (GASTA) to oscillate as its ‘seasonal variation’ (i.e. GASTA rises by 3.8°C from January to June and falls by 3.8°C from June to January during each year). Hence, the system is always seeking equilibrium but never achieves it.

    Such a varying system could be expected to exhibit forced oscillatory behaviour, and the ‘seasonal variation’ shows that it does. Importantly, some oscillations could be harmonic effects of the ‘seasonal variation’ and have periodicity of several years. Of course, such harmonic oscillation would be a process that – at least in principle – is capable of evaluation.

    However, there may be no process because the climate is a chaotic system. Therefore, the observed oscillations (ENSO, NAO, etc.) could be observation of the system seeking its chaotic attractor(s) in response to its seeking equilibrium in a constantly changing situation.

    Very importantly, there is an apparent ~900 year oscillation that caused the Roman Warm Period (RWP), then the Dark Age Cool Period (DACP), then the Medieval Warm Period (MWP), then the Little Ice Age (LIA), and the present warm period (PWP). All the observed rise of global temperature in the twentieth century could be recovery from the LIA that is similar to the recovery from the DACP to the MWP. And the ~900 year oscillation could be the chaotic climate system seeking its attractor(s). If so, then all global climate models and ‘attribution studies’ utilized by IPCC and CCSP are based on the false premise that there is a force or process causing climate to change when no such force or process exists.

    Richard

  12. On your diagram for the mythical May Island squirrels you should correct green trace to blue trace here ” …With a growth rate below 3 – 2.7 in this example (green trace) – stabilizes at 0.6xxx,…”.

    • Reply to goldminor ==> Thank you! I am blue/green colorblind and often call shades of these two colors by the wrong name. I had to open the image and blow-it up to discern the colors correctly. Much appreciated.

      At least you are paying attention!

  13. re : ~60 and ~900 year oscillation. I have derived both of these periodicities from a combination of the earth’s and lunar orbital periodicities. Naturally this would occurred if the temperatures are function of the tidal events, or simply it could be a ‘numerical construct’ of interference patterns .

  14. I’m just going to grumble that I think that “chaos” is the wrong term to apply to a situation which is rule-bound. But it is too late now. It has become the term of art.

    • I sympathise.

      I wrote something once on the 20th century discovery of various insoluble problems. And in fact that mathematics of the 20th century was more about proving what could not be proved, than anything else.

      Science is all about differential equations with respect to time. Sounds weird? Let me explain.

      Any theory is essentially a shorter way of describing what happens (in time) and, if its a useful theory, predicting what’s going to happen (in time), and the way that is done is to establish a differential equation of the quantities of interest, over time. So a simple law like F=ma is in fact a time differential of the relationship between force, mass and acceleration that can be integrated to give is positions and velocities in times in the future, where we know what the acceleration and masses are, and where those are constant.

      But what happens if the accelerations are not constant, that is the body in question might fall under the gravitational influence of some other body as its position takes it closer… This is a system that now has in a sense feedback,. Where it gets to depends on where it goes as well as where it starts.

      Its still deterministic, but its not soluble by simple maths.

      What I wanted to say is that we have classes of problems

      Easy= simple linear differential equations where not only the starting conditions are well known, but subsequent positions are dominated by terms that are , or are nearly, constant.

      Hard= complex non linear equations where even if the starting conditions are well known, subsequent positions are dominated by terms that are manifestly not constant.

      Impossible in practice= complex non linear equations whose actual shape is not well understood, and where even if the starting conditions are well known, subsequent positions are dominated by terms that are manifestly not constant. I suspect that’s where climate prediction is currently.

      Impossible in theory = complex non linear equations whose shape is well enough understood to calculate that the computational power to calculate where the system is going to end up requires a computer as big as the system under investigation. I.e. The earth’s climate is in fact the analogue computer that is running to tell us what the climate is going to be as soon as it happens ;-)

      None of these are ‘chaos’ though…
      =================================================================================
      …philosophical aside…
      An interesting question to ask is ‘How does the worldview of the third rate climate scientist or AHGW protagonist differ from the world view of the first rate sceptic’ ?

      And its actually possible to put a finger on this: Courtesy of Roger Scruton*, the warmist lacks pessimism. He believes, or purports to believe, that human minds can, given enough time (and of course taxpayer funding) solve any problem they are presented with,. And therefore we should just get on and solve them.

      The first rate minds who have gone neared to the bleeding edge of human knowledge, are aware that there are classes of problem we can prove are insoluble, certainly with known techniques, and possibly intrinsically.

      They are far more likley to be skeptical. They are more pessimistic. Because they understand the limitations.

      *A Cambridge philosopher ‘of the right’ as his leftish colleagues call him.

      • Your point about “pessimism” provides an interesting way to divide up the idea space. I need to give that some thought.

        In the meantime, I’ll succumb to the temptation to comment on the gap you’ve left between “easy” (linear time-invariant) and “hard” (non-linear time-variant). Perhaps there’s a “moderate” category (populated by, say, linear time-variant and/or non-linear time-invariant)?

  15. Don’t forget the Sun in with the ocean/atmosphere coupling. That is a thought I had as I read through to the end of your very clear post. Thank you for such an enlightening post as I was able to grasp a good bit of what is contained in it. I see that I have much more reading to do, but I feel ready for the task.

  16. Excellent work. One small point a roulette wheel that has 0 to 36 slots is actually 37 attractors. The odds on a single number are 36:1 which gives the house it’s edge. The 00 is 38 giving the house an extra advantage.

    • Apologies. My reading comprehension completely failed my this morning. That is exactly what the article says.

    • Reply to Clovis ==> You are right, however, about the purpose of the extra 00 in a ‘Vegas wheel.

      The probability of getting any one number is 1 out of 38 in Vegas. The payout for a $1 bet correctly picking a correct number, 00, 0 or 1-36, is $36. Thius, the House has an automatic long-term $2 advantage. In Europe, where there is only one 0, thus 37 possible results, the House advantage is only $1. Vegas doubled its take with the addition of the 00.

    • I read the first one, didn’t bother with the second; the first isn’t worth the .pdf its printed on, rendering its implications equally meaningless.

      From the first paper:
      “The Hadley Centre of the UK Met Office maintains a detailed global surface temperature dataset from
      1850 to the present (Hadley Centre, 2013) (Hadcrut4, 2015). The Hadcrut temperature data format
      selected for this study are the monthly mean land and ocean surface temperatures from January 1850
      to September 2015 in ensembles of gridded squares. Three hundred such gridded squares are identified with 100 each in the Tropics (30o North to 30o South), the Northern Hemisphere (north of 30o N), and the Southern Hemisphere (south of 30o S). A structured sample of 12 squares are selected from each ensemble of 100 defined as grid numbers 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, and 100. The sampling provides a wide coverage of the entire ensemble while maintaining a balance between even and odd numbered grids. A total of 36 grid datasets are analyzed for evidence of persistence using the R/S analysis described by Hurst and Mandelbrot (Hurst, 1951) (Mandelbrot B. , 1972) and with due regard to methodological issues described by Granero et al and others (Granero, 2008). In total, 36 Hurst
      exponents are computed, 12 for each of the three regions of the globe. “
      (emphasis mine)

      In other words, of 300 available data sets, they used 36. Imagine the howl from sites such as WUWT if Trenberth & Mann, et al., threw something that blatantly pre-selected at us in defense of the CAGW argument.

      It’s entirely possible that the paper’s conclusion (“The implication of the finding is that the stochastic process of nature that generates surface temperature can create patterns that may appear to be cooling trends, warming trends, and plateaus at decadal and brief multi-decadal periods to which no cause can be ascribed. “) is utterly valid. However, the paper itself fails to prove that thanks to its egregious sampling failure.

      As our friend Willis might say: when they get around to doing all 300 available data sets, let me know; until they do this is just fun with numbers.

      • Reply to Smokey ==> I agree with your conclusion too: “It’s entirely possible that the paper’s conclusion … is utterly valid.” but am not sure I agree with your main complaint.

        He is testing for “evidence of persistence” (a special definition of persistence, I believe) and to do that, he takes a sample of the available data sets (the grids). Nothing wrong with that approach, as long as he confines his conclusions to the “data sets tested”.

        The conclusion that “the stochastic process of nature that generates surface temperature can create patterns that may appear to be cooling trends, warming trends, and plateaus at decadal and brief multi-decadal periods to which no cause can be ascribed.“ is something I agree with. I don’t think that temperature creates temperature. See my reply to Nick Stokes here.

        That and the fact that any nominally random time series will develop what appear to be short trends that are not actually the result of an identifiable cause (they will be just part of the randomness).

        In Chaotic System, though, patterns can and do develop in what otherwise appears to be the random chaotic regions — some of these patterns, called Strange Attractors, are absolutely mind-blowing (to use a phrase from the 60’s, when they were first being explored).

      • @Kip and Chaam Jamal===> First off, I didn’t mean to be rude earlier with my criticism, so I apologize for that. In addition, I have no real issue with the conclusions reached by the paper, so my criticism was not intended to belittle that aspect of things either. Finally, I am not so hard-and-fast as the referenced Mr. (Dr?) Eisenbach, who wants the whole data set, all the time, every time; by contrast, I accept that there may be valid reasons for picking and choosing one’s data.

        Those important items being covered, my issue with this paper is that there are no reasons given for why only 12% of the available data was used to draw conclusions about the entire set. No methodology was given as to why the different sets were chosen for analysis, or else left out entirely, beyond a vague reference to maintaining a “a wide coverage of the entire ensemble” and “a balance between even and odd numbered grids.” Why having both even & odd sets made a difference was not explained, and a better coverage of the entire ensemble would have been gathered using the entire data set, so neither wipes away my reason for concern.

        Had the author so much as said “We only had time/resources to calculate a small part of the data, the initial results are interesting/encouraging, please give us more funding and we’ll do the rest of it,” I’d still have been alright with that. As it stands, it appears (no accusations nor insults intended) roughly as likely that the data were hand-picked to produce a result as it does that they were a genuinely representative sample of the entire data set. Because of this, the paper really cannot be taken seriously as anything other than an interesting exercise in mathematics, no matter how intuitively appealing the conclusion.

        The bottom line is that regardless of what this chunk of the data indicates, we still need the rest of it to know for sure, since we can’t know from this paper alone whether the data shown are truly representative of the entire set.

        I hope that makes my position a bit clearer. Again, my apologies for my rudeness earlier.

      • Reply to Smokey ==> De Nada.

        It is possible that the author should have said “The implication of the finding is that the stochastic process of nature that generates surface temperature can create regional patterns that may appear to be cooling trends, warming trends, and plateaus at decadal and brief multi-decadal periods to which no cause can be ascribed. “ or in some way indicated that they were not meaning to apply their result to global temperatures.

        Thanks for worrying about the possibility that you might have been considered rude. (I didn’t think so.)

    • Those papers’ abstracts are a little, well, abstract. Could you favor us with a paragraph or two that tell us what we will likely conclude if we invest the time to read both of those papers?

    • Reply to Chaam Jamal ==> Thank you for drawing our attention to these two papers. Do I assume correctly that these are your papers?

  17. Thanks for post. The technical discussions seem to be thinning out on this site, so your contribution is welcome.

    One minor ambiguity: the impression one might take from the discussion accompanying the second and third drawings (or the third and fourth, depending on whether you count the first’s repetition) is that a “kick” would always result in the system’s returning to the attractor value associated with the system’s r value. The equation suggests that won’t necessarily happen If the kick is big enough to make x exceed unity.

    • Stability analysis is based on the response to a small perturbation (Lyapunov stability), depending on the system of equations it is possible that a sufficiently large perturbation could leave the locally stable region of the phase space. There could be a surrounding stable limit cycle with a smaller stable focal region inside, in which case a sufficiently large perturbation would lead to a stable oscillation.

      • It’s certainly true that one could write phase-velocity equations descriptive of a system that behaves as you say. That the discrete-time relationship x_{n+1} = r * x_n * (1 – x_n) discussed by Mr. Hansen exhibits such behavior is not so clear.

      • Joe Born November 24, 2015 at 7:08 am
        It’s certainly true that one could write phase-velocity equations descriptive of a system that behaves as you say. That the discrete-time relationship x_{n+1} = r * x_n * (1 – x_n) discussed by Mr. Hansen exhibits such behavior is not so clear.

        Actually Bob May’s paper shows exactly that.

    • Reply to Joe and Phil ==> Ah, if we mix real life with constrained maths, we often see inconsistencies. In the Population Game, using this formula, one has to stay inside the unit limit. The system will/can never kick itself outside the limits. We are not allowed to either. Think of it as a confined space, with no “outside”.

      [ Albert Einstein had the same problem, but he just redefined reality to cover himself.]

    • Reply to Pointman ==> In the Chaos Theory sense, small errors introduced into the system change the initial state, which no matter how small, magnify over time so that later states may not only be numerically different, but the nature of the states may not even resemble the original. This was Lorenz’s big discovery and it shook him profoundly.

    • @Kip,

      All is chaos, a tectonic realisation to the deterministic mind-set of the 1960s scientist. God wasn’t even playing dice in any possibly predictable way …

      Pointman

      • Reply to Pointman ==> True, that. The thing that shocked the scientific world was that so many systems, virtually all non-linear dynamic systems (which may include nearly all real world dynamic systems) have chaotic regions….where behavior is entirely deterministic yet unpredictable and at the same time has areas that are strangely, wildly, beautifully ordered

        Ordered, deterministic, but not the Clockwork World they thought they had.

  18. Nit picking:

    I am not a fan of the term ‘stable’ in this context. I would prefer ‘bounded’

    It is obvious that cliamte is neither stable (and predictable) nor yet unstable (and totally unpredictable).

    Lorenz claims that chaos maths started when he discovered it. This is assuredly bunk. Ever since the three body problem was posited, plenty of people were aware of the general nature of chaotic systems. All except mathematicians, who woke up 400 years later and ‘discovered’ it.

    I suspect the reason for this was that ion general since you can’t predict chaotic systems reliably, developing detailed mathematical models was not a priority, and applied mathematicians applied themselves to linear problems that were soluble. Such studies as were done on chaotic systems were merely to determine ways to avoid them in practical situations.

    That apart, full applause for a decent job of tackling a very hard to explain subject.

    If chaos maths rather than ‘climate science’ were part of the O level syllabus..or equivalent in other nations ..

    The next question of course, is whether we can prove that climate is chaotic, and if so whether it can be predicted, or is even worth trying to predict.

    • Another intriguing comment.

      Unlike that of many commenters here, my purchase on the concept of chaos is tenuous at best. Perhaps that’s why it surprises me that a three-body system would be chaotic. For my benefit and that of any others here afflicted by the same limitation, could you explain how a three-body system qualifies as chaotic?

    • Reply to Leo Smith ==> Thank you for your support.

      There is a valid semantic argument — but as a long-time reader here, I know you are aware of my general use of simple practical language. I use Stability because of the nature of the single-value-attractor characteristic common to non-linear chaotic dynamic systems (outside of their chaotic regions) to return to their attractor after perturbation. So, this aspect may not simply be a matter of “bounding”.

      The graph of paleo-tempartures clearly illustrates that, as far as we can tell from this distance, the Earth’s climate is bounded…constrained to within certain values — not that we would be happy with another Ice Age, but we would not predict a total Ice Ball, or Fire Ball, Earth either — and seems or appears to be in a period-two oscillation between Ice Ages and Interglacials.

      The subject is more complicated than “stable” or “unpredictable”. So yours: “It is obvious that climate is neither stable (and predictable) nor yet unstable (and totally unpredictable).” is a bit off to one side. That is not the claim (either by myself nor by the IPCC).

      Acknowledging the climate as a “coupled non-linear chaotic system” does not mean that it is “stable” overall and it does not mean that it is unstable, totally unpredictable — it means that we should expect it to behave like a non-linear chaotic system.

      In this series of essays, I am attempting to lay out what some of these behaviors are — the behaviors common to non-linear chaotic dynamic systems — so that we can all have another scientific point from which to view what we can see as the climate system.

      One of these common characteristics common to non-linear chaotic systems are the periods of single-value attractors, which resist destabilization by returning to the attractor when perturbed.

      I posit that we might be seeing this in the last 10,000 yrs of surface temperature data.

      I hope this helps clarify the issue.

    • Reply to Leo Smith ==> re: “Lorenz claims that chaos maths started when he discovered it. This is assuredly bunk.”

      Read his book. Edward Lorenz was bemused by the fact that others wanted to call him the Father of Chaos. He never makes the claim — in fact, denies it rather forcefully. What is true is that it was his paper that got copied and circulated far outside its original field (which was meteorology) causing a rapid cascade of research on the topic — in widely divergent scientific areas.

      No one, especially Lorenz, thinks he was the first stumble on Chaos,

  19. I read Gleick’s CHAOS at the time and have never quite got over the power of the ideas in it. Often visual, software also allowed anybody to experiment with fractals and the creation of complexity from simple systems. The first thought I had upon reading this post was that the real take-home message from these theories was that simple systems can exhibit very complex behaviour.

    To equate the Earth’s climate with any of these theories is to go beyond there scope. The Earth is not a simple system, it is not even a combination of simple systems. This makes the IPCC statement quoted at the top of the post an utterly political one:

    “…we should recognise that we are dealing with a coupled non-linear chaotic system, and therefore that the long-term prediction of future climate states is not possible.”

    The statement is designed, I imagine, to equate climate with Chaos!

    It is possible that very complex systems, linear or otherwise in combination may produce simple behaviours and predictable outputs, such as temperature. I disagree, that broadly speaking, the long term temperature is unpredictable. I expect it will continue as it has for the last tens of thousands of years, in a very narrow range, punctuated as it has been in the past by extreme periods lasting similar amounts of time!!

    • I expect it will continue as it has for the last tens of thousands of years, in a very narrow range, punctuated as it has been in the past by extreme periods lasting similar amounts of time!!

      That is not inconsistent with : long-term prediction of future climate states is not possible.

      Let me explain. Climate has over millions of years varied enormously , but within a certain range. It has never since the earth first cooled down a bit, gone above 100C, and rarely gone below 0C as an average value across the whole globe.

      That is actually quite a narrow range in geological terms, but its a hopelessly wide range for human civilisation’s purposes. To say the climate will on average be 9°C ± 5 degrees is probably accurate for at least the next thousand years, but that could encompass a complete change in the socio-economic patterns of human behaviour, as well as a sea level change of tens of meters.

      The whole point (allegedly) of the IPCC is to assume the worst, get some figures for where the climate is headed and then offer advice to governments as to what to do. ± 5° C simply isn’t good enough to say more than ‘well just be prepared for almost anything from a plague of mosquitoes, to being underwater and living in a new coral reef, or living in an igloo’

      That is, in the context of what the IPCC is supposed to be about, they need it more accurate than that, and it is their conceit and their folly to think that they can be, or their deceit and their betrayal to pretend that they can, according to how sceptical you feel ;-)

      The whole thing hinges on ‘very narrow’: ‘very narrow’ may not be narrow enough.

    • That is how I have come to see it in my short time of following the story. The IPCC and those who adhere to that line of reasoning are correct in that they will never be able to correctly predict as they have blinded themselves to that which is right in front of their eyes.

    • Reply to SWB ==> The IPCC statement stems from Lorenz’s original discovery in the 1960’s that weather prediction is highly sensitive to initial conditions — a large part of what later became a major foundational element of Chaos Theory (not really a theory, and not really chaos…but what can you do at this later stage in the game?) Climate (cumulative weather) is subject as well to being highly sensitive to initial conditions.

      The atmosphere, as a dynamic system, is a non-linear dynamic system — composed of complicated, non-linear fluid dynamics and heat-transfer dynamics — thus correctly called (using the Chaos Theory sense) a “chaotic system”.

      The oceans, as a dynamic system, are a non-linear dynamic system — composed of complicated, non-linear fluid dynamics and heat-transfer dynamics — thus correctly called (using the Chaos Theory sense) a “chaotic system”.

      It is well-understood that the oceans and the atmosphere interact with one another in complicated ways and are understood thus to be “coupled” — changes in one bring about changes in the other.

      Wrapping that all together — the Climate can correctly be referred to as “a coupled nonlinear chaotic system”. BUT, Scott, the key is understanding that “chaotic” has a very special meaning when they say it — the meaning that is peculiar to Chaos Theory..

      I am not privy to the inside functioning of the IPCC, but I can’t imagine that they were pleased to be forced to admit this, as it means that they can’t predict future climate, which is what they want to do. It is not a dodge, but an unfortunate reality that they have been forced to admit. Despite the admission, they proceed to give all sorts of probably-nonsensical projections of future climate.

      • Reply to KH ==> I’m not sure we are disagreeing about anything here. We are more like ships passing in a fog though, than two ships in the night. I am a big fan of Lorenz and have a very sophisticated understanding of Chaos as I’ve had a long interest in these topics. In short and to quote, “The map is not the territory” a model of reality is not reality. This notion has been bitterly defended by a ragged band of artists, poets, philosophers (And the ocassional scientist ;-) since time immemorial. i.e. Plato or the Upanishads (c800 bc) for example. Lorenz’s insights came specifically from attempting to model weather using computers to calculate non-linear equations. Reducing the number of decimal places saved computer time but also produced completely different results. The sensitivity to initial conditions was a problem of computation not of reality. But as you said, you “will not be defending the idea that the climate system is a coupled non-linear chaotic system ” I won’t go any further here. Cheers SWB

    • It’s probably too inconvenient for Mr. Watts to provide, but the comment-entry arrangement at the JoNova site is worthy of emulation. In addition to permitting edits, it so numbers the comments as to make the reply hierarchy readily apparent.

    • Repply re Editing Comments ==> It is more complicated than that….

      There are economic issues (costs of hosting with WordPress, costs of features and special features, as I understand it). There are philosophical issues — does one allow “take-backs”? as in “How long after posting should editing be available? Minutes? Hours? Days?”.

      While I have “Happy Finger days” (on which my fingers type what they want, not necessarily what I want) , and it is embarrassing to look like a high-school drop-out based on nothing but common typos — (I make a lot of typos….) I think I’d rather see readers add a reply to their own comments with corrections….or apologies if they’ve “flown off the handle”.

      Note that Anthony has recently begun carefully documenting at the end of essays any and all edits made to the essay after-the-fact of original posting. All changes involve strike-outs that clearly show original words. There are reasons for that as well.

      • As to the editing, you don’t really have to allow takebacks. At JoNova there’s a view-WYSIWYG step: before you commit your comment you can review it as it will actually look.

        And I may be making this up, but I seem to remember a site where you have, say, five minutes to edit something before it’s indelible.

      • Reply to Joe ==> There is a private Comment Formatting add-on to Firefox that makes it possible to live with some of this — it includes a Preview function.

        The add-on and info is available here: http://climateaudit.org/ca-assistant/.

        It is easier than it look to install. Just a couple of steps.

    • Reply to SWB ==> Yes, that’s the way I think it should be done. Nothing wrong with a little humility — I could sure use some.

  20. Hey Kip,

    Did you ever wonder why the AR4 and AR5 notably ‘dropped’ the wording from the TAR ref. “…a coupled non-linear chaotic system… [which renders the prediction of long-term trends impossible]”?

    Did you by any chance attempt to reach the Lead Author(s) from TAR and AR4 WG1 to determine their intent in using (respectively in NOT using) that phrase?

    • Reply to Kurt ==> Well, they can’t repeat everything they’ve ever said in every report. They have not backed off from the fact though.

      I’m sure they are embarrassed by the reality — and don’t wish to keep bringing it up. They still stick to the point of “we don’t make predictions”…which are precluded by the chaotic nature of the climate.

      • Thanks, Kip.

        The quote on the INHERENT unpredictability of earth’s future climate from TAR (2001) is a very powerful one. It would truly be enlightening to read an interview of WG1 lead authors from the past three reports on this very question (including why it was left out of AR4 and AR5).

        The “we don’t make predictions” claim strikes me as daft. The ‘experts’ repeatedly claim that the upper and lower bounds of the global mean surface temperature trend are VERY robust; that long-term climate IS predictable, whereas short-term weather is not. Yet the DATA don’t seem to cooperate very well, n’est-ce pas?

        BTW, I’ve pushed for opting out of the Verified Commenter’s status over at Revkin’s blog (NYT Dot Earth). Perhaps others may wish to follow suit. IMHO, this will improve the conversation.

      • Reply to Kurt ==> As you know, the IPCC always calls them “projections”. They are based on obviously chaotic results of climate models — see the original “bloom” graphs of the 1000 runs of the models that produce those neat little graphs with spreading triangles representing the “means” of those runs which they call “upper and lower bounds.”. The actual individual model runs produce projections that could be anything from Fire Ball Earth to Ice Ball Earth. Then they “average” the chaotic results….I don’t know how to characterize this nonsensical approach. Dr. Brown at Duke has written repeated, very amusing, rants on this topic.

        As for Dot Earth….I inquired years ago into whether or not I could have an alternate sign-in not tied to my Verified Commenter status, so I could post comments that went to Andy Revkin for moderation, as I like to ask him questions. he doesn’t see the Verified comments normally. I was told it is one way or the other. It will be interesting to see what they come up with as the new system.

  21. If a system has two stable attractors, (like the Earth and its oscillation between glacial periods), what is the nature of the cause for switching between them? Is the event that causes a switch between one and the other the same concept as a “tipping point” that is lurking within the system itself and waiting to be tipped? Or is it caused by a major change randomly coming from outside the system?

    • T-Braun, in the case of the earth you have a variability in the orbit which can lead to changes, however there are also random internal changes which could be responsible. E.g. the blocking of the connection between the Pacific and Atlantic oceans by uplift, change in atmospheric circulation due to the uplift of the Himalayas, etc.

    • Reply to T-Braun ==> Good question!

      Generally speaking, a non-linear dynamic system, which is by its very nature prone to chaos, has built into it a time-factor. Ocean dynamics centered on the photosynthesis of plankton, for example, are timed by the Sun, the day length, the nigh-and-day cycle and by the seasons, a yearly cycle, and by various ocean circulation patterns with longer cycles. One would expect that photosynthesis of plankton system would be complex then, and it might display periodic behaviors on more than one scale.

      It is the nature of chaotic dynamic systems to be periodic in this way — with the time periods – one or more — depending on individual systems.

      Populations, for example, of terrestrial animals and plants, have a yearly cycle for obvious reasons related to the seasons and breeding cycles. When simple population systems (those not primarily driven by the predator-prey relationship, which as different dynamics, but is also chaotic) are “pushed” too hard — when their growth rate rises past 29…. or so, they become periodic, and the period would be based on the yearly breeding cycle. A squirrel plague one summer would drive current population down dramatically, and this would show up for the next few years as it does in the second population graph in the essay, with the blue trace, at year 50, with the population dragged way down, which then pops right back up, wobbles, and then settles down to the single-value attractor.

      A Tipping Point, however, is defined (for Climatology) in the Wiki as follows:

      “A climate tipping point is a somewhat ill-defined concept of a point when global climate changes from one stable state to another stable state, in a similar manner to a wine glass tipping over. After the tipping point has been passed, a transition to a new state occurs. The tipping event may be irreversible, comparable to wine spilling from the glass: standing up the glass will not put the wine back.”

      I believe the term to simply be a Bogey-Man — invented loosely based on Chaos Theory to frighten little children and politicians. The idea is that Mankind will somehow do something that will precipitate a “disastrous climate” — usually Fire Ball Earth. There is no indication in the paleo-record that Fire Ball Earth is an option. To the contrary, the opposite is true. Any perturbation is likely to move the system slightly but, with the exception of the return of an Ice Age (a long a slow process, I think), the system seeks a stable, well-bounded range.

      In the bifurcation diagram, you can clearly see that there are points, quite narrow points, at which the graph changes from a single-value attractor to a periodic-attractor, and then, in what is called a period-doubling cascade, rapidly transitions to a limited space-filling chaotic (deterministic but unpredicatble) region. In the diagram given, this occurs when the “r” reaches 3.6 or so.

      Many feel that the climate is in a long-term period-two state — Ice Ages and Interglacials, as you say. If this is the case, there is not a “Tipping Point” but rather the system “has always been headed there” to the point of a shift of regime. This is almost the same as “We Don’t Know What Causes the Shift”. We know it happens, we know it happens on a fairly regular basis.

      For our squirrels, a growth factor of 3.5 produces a wildly varying population, slamming on a yearly basis from 80% of the carrying capacity to 40% and back again — a doubling and halving . Nothing happens to “cause” it. Biologists come and rip their hair out trying to find out what is causing this terrible situation, slinging blame right and left. They somehow forget that population dynamics are non-linear…..

      I hope this helps.

  22. Thanks, Kip Hansen, for a very interesting essay.
    You point to an interesting breakpoint in the history of the IPCC; the missing chaos that used to make long-term predictions impossible. TAR is dead, FAR killed it.

  23. The true chaos here is that every year “adjustments” are made to past average temperature data since 1880, so that the gap between reality and the official numbers increases.

    In time, I expect the 1930’s US “dust bowl” will be remembered in the history books as the 1930’s “snow bowl”.

    • Reply to Richard Greene ==> Children will ask: Was the Dust Bowl a famous football game? … like the Rose Bowl? the Orange Bowl?

  24. Re: WUWT, Chaos & Climate Part 2, 11/23/15:

    Chaos & Climate is about this lead quotation from IPCC:

    “…we should recognise that we are dealing with a coupled non-linear chaotic system, and therefore that the long-term prediction of future climate states is not possible.”

    Part 2 corrects only the attribution. Part 1 said it was AR4 WG1. Now in Part 2 it is TAR Chap 14, Exec Summary. The significant thing that should have been corrected were IPCC’s conclusions about the climate system.

    As I wrote on 3/15/15 @ 10:19 am,

    What the IPCC, like many others, has long recognized is false. Linearity and chaos apply to models, not to the real world. Whether models are linear often depends merely on the scale factor* applied. A system is linear if f(ax+by) = af(x)+bf(y). Equation do not exist in the real world to apply the definition. Models and equations are strictly manmade.

    Chaos does not exist in the real world because the real world has no initial conditions to which it might be sensitive. Only models do. Climatologists and physicists alike often confuse the real world with their models.

    The ensuing discussion there did nothing to change the facts, or the definitions of the terms chaos and linearity, nor, apparently, to remedy the confusion between models of the real world and the real world they model held even by people who should be able trained in some scientific field and who should be able to grasp the problem. This is a matter of epistemology, but more importantly, it is a matter of scientific literacy, a skill sorely lacking among voters.

    On an obviously higher plane, Kip Hansen, for example, defended his essay on 3/15/2015 writing this:

    Turbulence in fluid flows of all kinds, including the atmosphere. Heat transfer through and between materials. Everyday population dynamics. Passage of radiant energy through a translucent medium (the atmosphere). All nonlinear dynamical systems, and subject to all the behaviors of nonlinearity. @ 11:25 am

    and

    There are so many real world examples of these chaotic behaviors in natural systems that I find your continuing assertions to the contrary difficult to understand.

    A couple of examples might help Mr. Hansen in his quest for understanding. *An example of model dependence on scale factor, mentioned above, is radiation transfer. At the molecular level, it is nonlinear. But at the heat transfer level, it is linear, analogous to Ohm’s law. In fact, the entire climate system can be modeled linearly as heat transfer rather than radiation transfer, and with better – with success. Another example of model dependency is the flux of CO2 between the atmosphere and the surface ocean. According to Henry’s Law, uptake by the surface layer is linear in atmospheric CO2 concentration. Outgassing, however, is nonlinear, depending on the reciprocal of atmospheric CO2 concentration. IPCC doesn’t use Henry’s Law, and when it appeared in its experiments on the Revelle Factor, IPCC concealed its results. The real world flux is neither linear nor nonlinear; neither nonlinear nor chaotic. And this holds with the world flux deleted.

    If Kip Hansen wants to prevail with his argument that chaos and nonlinearity exist in the real world, he needs to accomplish these three things:

    1. Quote a reasonably acceptable definition of linearity (and its absence, nonlinearity),

    2. Quote a reasonably acceptable definition of chaos, and

    3. Choose any example of a real world process and show that it fits those definitions without resorting to any scientific model for the process.

    As things now stand, each point is an independent, glaring omission in his continuing essay.

    P.S.: The origin of IPCC’s claims of chaos and nonlinearity in real world climate is its excuse for its inability to model climate successfully, meaning not that it failed to be (1) peer-reviewed, (2) published in a certified journal, and (3) recognized in a certified consensus, but that its models are now proving invalid in making better-than-chance predictions, i.e., equilibrium climate sensitivity today measures at or below IPCC’s 5% confidence level, extrapolated.

    • Reply to Jeff Glassman ==> I quote from the Author’s Comment Policy:

      I will not be defending the idea that the climate system is a coupled non-linear chaotic system – it is simply too well established and supported by the physical actuality. Those who wish to rail against this fact can have their say here, but I will not be replying. It would be interesting to read a well presented essay from those who hold such an opinion as to why they do so and I’m sure Anthony would post such an essay.

  25. Chaos is a description of a system or process that is incompletely or insufficiently characterized and unwieldy, thus preventing or degrading accurate predictions of its future states and paths. It is the implicit acknowledgement of people’s limited skill and knowledge that motivated establishment of a [limited] frame-based scientific philosophy and methods. A chaotic system can only be reasonably described by feature envelopes and estimated with stochastic methods within progressively limited frames of reference in semi-stable systems and processes.

    • Not correct. Chaotic systems are often quite simple, as the mathematical example given shows. What holds chaotic systems apart is, first, that they cannot be described with a simple time equation and, second, that they are extremely susceptible to “initial conditions”.

      For example, given 4 or bodies in space, (I can’t remember if the 3 body problem has been solved, but I know that the 4-body cannot be) it is not even theoretically possible to derive an equation to predict one body’s path. The only way to predict trajectories is through numerical simulation, and even then you can only predict a certain distance into the future. The bit errors in your computation, and your limits of knowledge of the exact positions and velocities involved, will cause your model to eventually diverge from reality.

  26. Left side of graph has intervals of 200,000 years, while the right side is 5000 years, so you are comparing apples and oranges in terms of periodicity and “stability.”

    • I have been thinking that Ren’s sudden stratospheric warmings (SSWs) may be a ‘butterfly’ in short to medium term seasonal weather. They seem to have the characteristics, If so, this leads to an observation that: Chaos, like magic, is just another name for things we do not yet know, and cannot predict. But if Piers Corbyn can accurately predict this solar effect, who can say how much we may advance? If we study the right things for a change, that is.

  27. Reply to Barry ==> From the text below the graphic:

    “The left-hand panel shows the period-two behavior of Ice Ages and Interglacials. (Note: the time scale changes at the panel break. Earlier panels of the original image have been cropped out. There are questions as to whether the vertical alignment of the first, left-hand, panel is properly represented and/or supported by data.)”

    In other words, this is already explained.

    Re-read that portion and see if you can see the point I am making. The graphic shows incredible stability over the last 10,000 years and over the longer term, 1 million years, shows the period-two period. We are in one of the Interglacials — which started about 20,000 years ago.

  28. Kip, I hope this comment is not too late for you to see and respond. I believe it is important to examine the second part of what the IPCC said in the executive summary for AR4. The complete quote is here:

    “In sum, a strategy must recognise what is possible. In climate research and modelling, we should recognise that we are dealing with a coupled non-linear chaotic system, and therefore that the long-term prediction of future climate states is not possible. The most we can expect to achieve is the prediction of the probability distribution of the system’s future possible states by the generation of ensembles of model solutions. This reduces climate change to the discernment of significant differences in the statistics of such ensembles. The generation of such model ensembles will require the dedication of greatly increased computer resources and the application of new methods of model diagnosis. Addressing adequately the statistical nature of climate is computationally intensive, but such statistical information is essential.”

    The issue then focuses more precisely on the use of model ensembles to predict future climate states, AND is the IPCC doing this in a scientifically sound manner? Dr. Robert Brown (Duke University) has some thoughtful critiques of the IPCC methods. For example, the ensemble result from multiple models with different physics can’t be anything meaningful. My personal opinion is that even with the ‘perfect’ model we would still need to use non-parametric statistical analysis with each model realization (simulation) being equally likely.

    Thoughts?

    • Reply to dougbadgero ==> Never too late — I usually leave my email address in a Post Script in comments to any essay I write — just so those reading later and having a question can bring it to my attention.

      You are right — understanding the full context of a statement is very important. In fact — if not in word — the IPCC declares that it recognizes that the future state of the climate can not e predicted due to the very nature of climate as a couple non-linear chaotic dynamic system and then immediate preceed to explain how they think they can get around that and predict it anyway.

      i am impressed that you have read Dr. Brown’s thoughts on this topic — some of them are quite amusing. I explain the problem this way:

      ” They (projections) are based on obviously chaotic results of climate models — see the original “bloom” graphs of the 1000 runs of the models that produce those neat little graphs with spreading triangles representing the “means” of those runs which they call “upper and lower bounds.”. The actual individual model runs produce projections that could be anything from Fire Ball Earth to Ice Ball Earth. Then they “average” the chaotic results….I don’t know how to characterize this nonsensical approach.”

      Running a thousand or a million runs of a chaotic system (when in a known chaotic region) produces a million different results that are literally all over the map for the system. Averaging those chaotic results does not, can not, will not, can’t even be thought to possibly, produce a valid prediction of future states. The full set of runs does produce an idea of the possible breath of possible results, the space filled by the chaotic system, but the mean — by any method or measure — can not be thought to be more probable than any of the extreme values — by their very nature they are unpredictable.

      If you haven’t already, read his comments to my chaos first essay..

      This is how Dr Brown puts it in a later post:

      “…. current climate models, for all of their computational complexity and enormous size and expense, are still no more than toys, countless orders of magnitude away from the integration scale where we might have some reasonable hope of success. They are being used with gay abandon to generate countless climate trajectories, none of which particularly resemble the climate, and then they are averaged in ways that are an absolute statistical obscenity as if the linearized average of a Feigenbaum tree of chaotic behavior is somehow a good predictor of the behavior of a chaotic system!

      This isn’t just dumb, it is beyond dumb. It is literally betraying the roots of the entire discipline for manna.”

  29. Thank you for the reply. I went back to your original essay and read rgb comments. I had seen many of his comments here but not on that thread. Very interesting, notably some of the physics the models obviously get wrong, e.g. Model variance versus observations.

  30. Ahaaaag, “coupled non-linear chaotic system” is just gobbeldespeak for “it’s complicated”. As you say, every step is completely deterministic, but the only reason we can’t predict the result is that we are unworthy… so far.

    • Reply to Brian H ==> Well, that is an interesting question, though I suspect you meant it rhetorically.

      What types of things *can* be predicted?

      Science is about investigating reality. Part of this effort is to form hypotheses — guesses — as what what the relationship between things, or the causes of things, might be or have been. To be successful, your hypothesis must have predictive power. Often, a hypothesis is tested against the past — if my hypothesis is correct, then we should find evidence that a very large asteroid struck the Earth about 65 million years ago. That is “predicting” the past.

      One of the problems pointed out about the current crop of climate computer models is that they are not very good at predicting past climate (when run “in reverse”) thus doubt is raised about their ability to project future climates. [Note: this ignores that IPCC’s own stand that future climate states cannot be predicted.]

      Predictions can be for the present as well (in the sense of hypotheses) — some intelligence agencies have predicted that the Syrian refugees contain an organized and planned sub-set of ISIS 5th Columnists. I believe this will be found to have been true.

      So, in the loose sense used in Science for the testing of hypotheses, predict and future need not always be used together, nor does one require the other.

      Thanks for asking, interesting idea.

  31. g-sperm;
    Maybe some of the crucial determinant data is below the level of even theoretically feasible detection!

  32. POSTSCRIPT:

    Thanks to all who took the time to read this essay on Chaos and Stability.

    I appreciate the input and questions from all of you readers here in the Comments Section. A lot of good discussion, some good questions and some interesting side issues have made the Comments worthwhile. I hope you enjoyed participating as we shared knowledge and opinions, as we learned from one another.

    My thanks particularly to those who joined in on the question of:

    “… why I say in the last paragraph before the Take Home Messages: “made up of at least two closely-coupled non-linear dynamic systems”. Might there be more than two? If so, what might the additional systems be?”

    Here’s a summary of the answers received:

    lithosphere noun, Geology.
    1. the solid portion of the earth (distinguished from atmosphere, hydrosphere ).
    2. the crust and upper mantle of the earth

    *****

    hydrosphere noun
    all the waters on the earth’s surface, such as lakes and seas

    *****

    biosphere noun
    the regions of the surface, atmosphere, and hydrosphere of the earth occupied by living organisms.

    *****

    cosmosphere noun
    the material universe

    *****

    heliosphere noun, Astronomy
    the region of space, encompassing the solar system, in which the solar wind has a significant influence.

    *****

    The Sun

    *****

    Solar System Orbital forces including the Earth’s Moon (indirectly suggested in a discussion of the tides affecting climate)

    *****

    How many of these suggestions are themselves non-linear chaotic dynamic systems, I can’t say off-hand, but all of them certainly have influence on the Earth’s overall climate system.

    The one system that I had in mind when I made the statement was:

    The Lithosphere, considered as the solid portion of the earth (distinguished from atmosphere, hydrosphere). Or if one considers the lithosphere to be only the crust and upper mantle, then the Lithosphere and Earth’s Core together. The dynamics of liquid-rock and heat flow are certainly non-linear. The core creates heat through radioactive decay (and still possesses residual heat from planetary accretion, which accounts for about 20% of the total) and that heat flows and is passed to the surface and thus, eventually, out to space. On its way, the heat warms both oceans and atmosphere. We know that the Earth’s lithosphere/core directly affects ocean temperatures where volcanism occurs under the sea (such as off the West Antarctic Peninsula), through undersea vents of hot water and steam, CO2 and methane vents, and direct general heating of the ocean from the bottom everywhere. Such uneven heating must affect ocean currents and create turbulences. Obviously, heat from the core adds to the oceans’ total heat content, and we have no idea how much it adds or whether the additions are even over time or uneven. (The same could be said of the lithosphere/core and the atmosphere, at least to some degree.)

    If this is true, and it seems obvious to me, the oceans are coupled to both the atmosphere and the lithosphere/Earth-core, one above and one below, making three coupled non-linear chaotic systems.

    If nothing else, it is an interesting idea that brings up interesting questions. (If this stimulates any further discussion, I’ll try to check in.)

    As always, if you still have some burning question that will keep you up at night, or have a comment that you just can’t hold back, feel free to post it below (though I can’t guarantee I’ll see at any particular time) or email it to me at my first name at the domain i4 decimal net , I’m always glad to hear from folks who bother to read the things I write.

    Thanks again,

    Kip Hansen

    • Reply to Dave Fair ==> See the point in the post script about the lithosphere/Earth’s-core being a possible third item in the “coupled non-linear chaotic system” that is Earth’s climate.

    • Reply to Dave Fair ==> That is a good example of how the solid part of the Earth, which is a ball of hot rock, some of which is liquid hot rock, can affect the climate — in the West Antarctic Peninsula, at least regionally. All the heat that “leaks” out into the oceans at hot spots, volcanoes, steam vents, etc must be doing something to the climate….it is the “other” source of energy/heat input into the climate system. The Sun, of course, is the one usually considered — often considered to be the “only” source.

  33. Thank you, Captain (from an ex-drill Sgt.).

    As pure speculation, by an ex-engineer, I wonder if ENSO, AMO, PDO, etc. are simply nature’s “one-finger-salute” to misuse of mathematics by climate modelers?

    I wish you a pleasant Thanksgiving.

    Dave

    • Reply to Dave Fair ==> Some of these circulations and oscillations were first “predicted” in experiments by Dave Fultz — the famous dishpan experiments. See here.

      These experiments are the original inspiration for my thoughts about the heating of the oceans from the bottom and heating them in undersea volcanic zones — like a Bunsen burner under the dish pan.

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