Guest Essay by Kip Hansen
“…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
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.
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 , 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 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.
Figure 1: “Bifurcation diagram of the logistic map x → r 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 — x → r x (1 – x) — with a beginning “r” of 2.8, and an initial state value of 0.2, this is what we find:
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:
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.
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.
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.]
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.
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:
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:
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:
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.
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.
- 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.
- 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.
- 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.
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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.
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[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]