Hello George,

That's the question... It seems to me more of a question of prejudice for "deterministic", rather than any proof the climate is deterministic, i.e. it is wishful thinking, but ultimately a moot point, because in the end the climate is at rock-bottom random.

There are different scales of climate (weather), Luisiana. Some of them are stochastic, but some are predictable.

Eugene,

Not true... On smaller time scales, millennium, the climate is a stationary stochastic process, on larger scales, say, 400,000 year timescale, it is cyclostationary stochastic process and larger still, millions of years, the climate is a non-stationary stochastic process; but, on no matter what timescale, the climate remains a random process.

But in very small time scale (weather) events are evidently predictable.

Not all, but most.

Weather is predictable out to about five days; beyond that...to no avail. Most scientists have admitted that no weather model does any better than a random model, hence, why bother?

Ah, Theophanes, herein lay the conundrum. For example, Earthquakes are inherently random, but one may easily argue that they are deterministic, based upon the stresses and strains of the tectonic plates, moreover, if one only knew all the pressures and movements, one could predict--with infinite accuracy--the next Earthquake; yet, we do not have this information to enable prediction, therefore, this process must remain a random process. 

Now, as for quantum systems, since the climate is comprised of quantum particles, including atoms and molecules, then climate ultimately rests upon what you admit to be a random process. It is true that in the aggregate the system approaches a an average, etc; nevertheless, one must employ statistics and not deterministic models for the climate.

Another example: particles in a box, the molecules in the room you are in have a distribution of velocities and if you knew with perfect knowledge all this information, then you could predict a future state--the very definition of deterministic. But, you are not privy to such knowledge, therefore, must resort to a statistical description. Moreover, why bother building a model that keeps track of some astronomical number of molecules, when you can simply use statistics to describe--with great accuracy--the eventual future state.

If one finally surrenders to the fact climate is ultimately random, then one can at most declare the mean and standard deviation...and that is all. 

Determenistic chaos, Theophanes... consequence of non-linearity (philosophically self-referencing).

Turbulence is also stochastic process, Luisiana, but some of it's laws can be deduced.

Microscopic linearity is illusion, Theophanes. All real problems (Fock, etc) are nonlinear. In linear theory nothing happens. It is only base, like Cartesian coordinates in geometry.

OK, a bit of a divergence from the question, but in the same vein; nevertheless, the same conundrum, namely, whether the system be defined random or chaotic (deterministic), yet, ultimately, predictability fails in either case, except for the celebrated mean and std, then why bother insisting on a chaotic description if predictability eludes capture?

So, climate may be envisioned 'deterministic', yet the ability to capture the impulse response function eludes our capture, then why not just resort to a statistical description? The answer: money (PERIOD). If you claim it chaotic and you need a supercomputer, models, etc, then we have a career; although, for the average taxpayer, many of which could care the least about climate on millennial scales, this all becomes a severe drain on their resources...potentially threatening the future of themselves and their children. Be it as it may, what about the misappropriation of these funds for the sake of science? If all this money is thrown at a hopeless endeavor, then what hope of addressing the host of other serious problems faced?

...and I agree with Eugene's statement regarding nonlinearity, which leads to intractability of many problems, hence, it is more efficient to resort to statistical descriptions.

Chaotic system would be a deterministic system which produces deterministic states having random-like distribution in space and time. By appearance may not be well distinguishable from a stochastic system in the same manner as a  pseudo random number generators from true random number generators.

Chaotic model by no means eliminates randomness from the nature or the lack of precisely known initial and boundary conditions, which add on the same dose of complexity as to any non-chaotic system.

People like catch words to describe reality. With just one word you can never be non-ambiguous and accurate.

We have models on different level of fidelity, different resolution and we should always take into account what matters on each level and we should expect side effects of unavoidable simplifications.

All right, but the question remains: what practical difference lay between stochastic and chaotic systems? If one behaves randomly and the other appears random, then I say, "Nothing is different." The issue is time, labour and expense to model a random system as if it were deterministic, say, the climate, which is random in all regards; therefore, a waste of time and one only befuddles themselves. 

 stochastic processes may not be chaotic. Chaotic system also can be deterministic or stochastic.

Please, elaborate, Lejiang.

Hello Lejiang, you'll need to explain that statement for me as well...

Hi,

to distinguish between stochastic and chaotic climate dynamics is of fundamental importance, I think. The apparent question behind: is the climate system in a quasi-equilibrium state around which it just fluctuates, or is it in an excited dynamical regime apart from equilibrium (and if so: how far from equilibrium)? There is no question (to me) that there are stochastic elements in the system's motion. As far as I could infer as yet from modelling and real-data analysis studies (cf. two of the featured papers of my RG account), the system passes a couple of bifurcations during its seasonal march. These are by definition situations that bear high sensitivity with respect to each individual trajectory the system follows. That is, climate dynamics clearly bears stochastic aspects (recall the irregular wander of the Southern Oscillation, for example). Topology and geometry in the back of its motion in phase space, however, may nevertheless exhibit an organized system. Indirect indication for the latter is borne in synchronous motions galore found across the instrumental climate record. A basically stochastic system may hardly exhibit a similar degree of synchronous motion (external and internal synchrony). And last not least, the small GCM that I have run shows a route to chaos in its intraseasonal monsoon activity (difficult to treat computationally, but nevertheless an emergent structure that shows the observed dynamics in boreal summer in a qualitatively interesting manner, including phase space topology and geometry in the back).

Hello Peter,

  Thank you for your informed and insightful contribution...it appears you are well versed in the matter. I also agree, there are many patterns within the data, either temperature or otherwise; but, what is of concern in the question of pragmatic versus academic, where the latter may give endless hours of delight, but the former provides the user with something real, practical and useful.   

With that said, even if the climate system is totally deterministic, of which, I must admit is true, otherwise it would be a macroscopic system devoid of any underlying regular, rational cause...nevertheless, despite this fact, it is ultimately an unpredictable system, behaving as a self-stationary stochastic system; moreover, this has similarity with many other complex systems, say, particles in a box, etc. 

I too find many hidden patterns within paleoclimate data, but what is more practical and efficient is to accept that the overall characteristic of the data succumbs to efficient statistics, hence, allows both global and local descriptions--couched in statistics, of course--and permits prediction in the future by virtue of stationary stochastic processes and their descriptors. In other words, a few statistical descriptors and we're done.

Of course, the question of CO2 emission and potentially forcing the Earth's climate into some catastrophic unstable node...well, the data does not support this perspective, also, nor does the physics. 

Hi Luisiana,

interesting points of discussion, indeed - give me two days for my answer, please (shops are closing soon, and tomorrow is the "day of labour" - where they remain closed, ironically).

Best, Peter

By all means...I'll await for your answer. I think the discussion is going more towards what I am seeking to answer, specifically, the age-old issue in science of deterministic versus complexity, where the latter must be described statistically, which many would describe as being a form of defeat; nevertheless, as Laplace has stated, "(Of course, not a quote, a paraphrase) Only the mind of an infinitely intelligent being could keep track of so many trajectories."

Lorentz system is a determinstic system wth three parameters and can generate chaoes on the condition of certain parameter. A determinictic system with less than three parameter usually can not produce chaoes, which has been proven by some researchers. In fact there is hardly deterministic system in climate and ocean researches. We usually add a Brown process to a deterministic system to study the climate and ocean processes. If you are interested in Nonlinear dynaimical system, Please read the book "Nonlinear climate dynamics and Nonlinear physical oceangraphy by Henk . Thank you

Lejiang,

The point of the discussion is, "What practical difference lay between stochastic and chaotic systems?" In other words, regardless if the climate or weather is actually chaotic or not, if measurements are indistinguishable from a random set, then statistics are all you need to model the climate, hence, all attempts to build complex climate models are effete, fruitless and unnecessary. It is far more efficient and sufficient to use statistics to describe, including predicting the future behavior of the climate. 

Leonard, I absolutely disagree! Take a heartbeat, for example, it is a chaotic oscillator, but affords no means of predicting the shape, time and etc, of the next series of heartbeats. So, chaotic systems are rational and theoretically deterministic, but are practically speaking, just the same as stochastic systems, specifically, unpredictable. 

Thus, why waste time claiming the system is chaotic, therefore, deterministic, therefore, can succumb to prediction, yet it is not? I find the label of chaos more an egotistical requirement of scientist, rather than for anything meaningful or practical. 

The biggest structure difference between chaotic and stochsticsystems is that chaos have the self-similarity sturcture ( fractal).

Ah, finally! That is a characteristic of some chaotic systems, self-similarity; but, stochastic systems enjoy a type of self-similarity as well, especially, if they are ergodic... Another example: " Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales." The quote is taken from Wikipedia regarding self-similarity.

So, in short, self-similarity does not exclude stochastic systems from chaotic systems; moreover, the question still goes unanswered, "What practical difference lay between stochastic and chaotic systems?"

I think that realisations of chaotic systems trajectories definitly behave like stochastic processes. But than you can ask yourself: is the systems of stochastic differential equations (like Langevine-type) are chaotic systems? Seems it depends on your test. For my test chaotic systems contain the way to chaos in their structure like Lorenz system. But you can add aditional stochastic force to Lorenz model and the system will remain chaotic.  In linear Langevine equation all stochastisity originated from additional stochastic term. If you will сonsider this term like the output of chaotic system then the whole system (friction plus noise source) will be chaotic system. That s for my test 

Chaotic systems may be the antithesis of stationary stochastic series. Think turbulence.

You can use conventional statistical techniques to analyzs stationary stochastic processes. But as the attached paper shows there are many problems in analyzing time series, even those that are more stable than chaotic processes.

http://forecasters.org/pdfs/IEEE_TNN_2008.pdf

The stationary stochastic process is mean-reverting, as is the deterministic trend process, at least in the sense of reverting to a local mean. Think seasonality.

Chaotic and other non-stationary processes such as the random walk are not mean-reverting,

Frederick,

A quote from the paper you provided, "While the results do not give a clear-cut universal answer to the previous question, they do provide insights on the modeling issues for trend time–series forecasting." In other words, none of the methods used did markedly better than any of the others, hence, it really makes no difference what you a priori decide what the underlying nature of the data is, whether it be chaotic, stochastic, etc...

Hello Lenny,

Point well noted and repeated. Stochastic systems are indistinguishable from chaotic systems, except for academic semantics.

The question is, "What practical difference lay between stochastic and chaotic systems?"

These are good points, Lenny, where the struggle many scientist are having describing a stochastic system as "deterministic" are well pointed out by such nomenclature as 'aleatoric'. That term, aleoatoric, is perfect to describe something you so badly want to be deterministic, but grudgingly admit there is something 'unpredictable' about the system. 

Regardless, in the case for climate, scientists have switched from describing the system as stochastic to being chaotic, where the latter term implies the climate system is susceptible to mathematical description in the classic sense. All the models built so far have only confounded the users, muddied the issue of what is climate change and cost taxpayers a mountain of money. That latter issue seems to be the real cause for the chase--if you get my drift. 

Good show... I want to second the motion on Ernst Mach's quote and sentiment regarding determinism and the inability of Mankind to discern what is or is not such, in fact, it goes directly to the point of the question at hand, which asks what "practical difference lay between stochastic and chaotic systems?" 

Now, as for weather and what not...let's not squabble over whether or not they are chaotic in the academic sense; but, let us agree to recognize that scientist are claiming the climate to be a chaotic system for one express purpose: to imply they are able to crack the code and provide to the community at large a model that would accurately predict the climate of some future date. That claim--I declare--to be absolutely and patently wrong! There are many reasons, but in the context of this discussion, the issue is that there is no practical difference between chaotic and stochastic systems, not from the analysis point of view, and we must accept the best we are ever going to do for climate predictions are those made from statistical descriptors.

Hello George, thanks for vote! Yes, next year will be another story, much like the same in the medical field, where statistics are so ill understood and applied that one might as well consider taking any drug akin to betting on the roulette, especially, with regard to bad side-effects, as an example, Ritalin and men producing lactating breast. We'll suffer this back and forth for a little while longer, but the general public is going to grow tired of the noise from the scientific community regarding this year's climate predictions; as a consequence, many in the public are going to lose their faith in science, equally, the politicians that attempted to manipulate public opinion, plus, any hope of solving the societal problem of waste and pollution.

All right, I've vented enough...I'll sit back and listen to the opinion of others...

Re: "It seems in the history of climate science that most scientists spoke of the climate as being stochastic, until about the mid 1980s, where a shift occurred and the climate was more described as being chaotic."

Recall the initial question, which sought to distinguish between stationary stochastic processes and chaotic systems. A process is by definition a component part of a system. The first scientific problem is to distinguish between a system and its components. 

We can agree that the climate system of Earth taken as a whole is chaotic, and as you imply, the IPCC is on record as having defined the climate system as chaotic and inherently unpredictable.

URL http://www.ipcc.ch/ipccreports/tar/wg1/504.htm

However, even if the climate system were not chaotic, the system is so complex that we cannot model it as a deterministic system. Professor Essex of the Department of Applied Mathematics, U. Western Ontario, explains why in a lucid video lecture. "Believing Six Impossible Things before Breakfast, and Climate Models"". Christopher Essex, Ph.D. URL: https://www.youtube.com/watch?v=hvhipLNeda4

By climate models, Dr Essex is referring to the GCMs that the IPCC relies upon. These are system models, which synthesize what is known about the climate system.

We may approach climate processes analytically (rather than synthetically) and find that some variables of interest are not chaotic.

But we have to be more careful than some climatologists and most climate bloggers. If a hammer is our only tool, then everything we see will begin looking like a nail. In climate science the tool of choice is the arithmetic moving average, a hammer if there ever was one.

In climate science, the Keeling Curve is well-known to statisticians as an example of a variable with a deterministic trend.

URL: https://scripps.ucsd.edu/programs/keelingcurve/

(It's a mere coincidence that this question was asked soon after I chose the Keeling Curve for my project to learn the R programming language. URL: https://geoscienceenvironment.wordpress.com/)

The Keeling Curve has several interesting statistical properties that I propose to explore..The annual series appears at first sight to be perfectly regular, but it is not. The weekly series appears to be perfectly sinusoidal, but appearances may be deceptive there also. Nevertheless, we can make a few general statements to get started.

The Keeling Curve appears to be trend stationary. The series is mean reverting, at least locally. If we imagine the series with the trend removed, the mean may be constant, as well as the variance and auto-correlation, or nearly so.

This first step in classifying the nature of the generating process will guide in making the series stationary. estimate the trend and then subtract the trend from the series. The result is a stationary stochastic series with constant mean, variance and auto-correlation over time.

Such a series can be analyzed using conventional statistical tools. In effect, we use a screwdriver to insert a screw.

However, if we transform the series using first differences, we will be using a hammer to turn the screw. Transforming by taking first differences is appropriate for series that are generated by a non-stationary stochastic process (such as a random walk).

In lectures, some statisticians cite the global temperature series as a random walk and then show that taking first differences produces a stationary series.

(Granger and Engle got a Nobel Prize for demonstrating this approach that tames a random walk with or without drift. (Think: drunkard walking with a dog on an elastic leash. Both the drunkard and the dog follow a random walk, but there is a relationship as loose as the leash is elastic.)

The next article in my blog about the Mauna Loa CO2 series will be estimating the missing values using a method that is determined by the nature of the generating process.

The practice of taking of monthly and annual means may not be appropriate for data with seasonality. In a later article in my blog, I plan to explore this question in relation to the Mauna Loa CO2 series. However, I will touch upon it in the next article in relation to filling in the missing observations.

Finally, I have experience in both physical and socioeconomic synthetic modeling. My results never enabled me to predict anything. But the models were indeed useful as learning tools.

In 1966 or so, I built a mostly deterministic model of a "postman's walk" for the UK Post Office using data collected by Post Office staff. The model was useful mainly for identifying errors in the data collection and the fact that one postman occasionally stopped for a cup of tea with a widowed pensioner on his route.

I love the story of the postman walk... I think the climate can be spoken of in the same vein, i.e. drunk and someone that just does not give a hoot. It is marching along, the climate, regardless of man's industry or activity. The CO_2 concentrations are definitely being affected by our activity, but the overall impact to the climate is vanishingly small and not worth noting.

Consider hierarchical systems, Luisiana, where flow of information from one level to another is limited and directied. What is chaotic on one level, may be deterministic on higher level. It seams to me that such is a life, Nature and other physical systems. The question about Mind (which level, hierarchical or not) remains.

Regards,

Eugene.

Eugene, 

A very good perspective to view the question from information theory. Although hierarchical structures are interesting in themselves, especially, the interconnections and flow of information; even though this be interesting, it is not necessary to contemplate such complexity to answer the question under this post. It would sound something like this: regardless if the system is chaotic or stochastic, from the stance of signal analysis, that is, the output or reading of the system, both versions appear similar in behavior and character. In some sense, the information level is very high in such systems, for the entropy is very large, hence, the system presents a high degree of unpredictability. 

For us all is information, Luisiana. We have nothing else. Again, Mind is the question.

I see in RG some attempts to reduce all to information. For example, Sergey Shevchenko's "Information as absolute". Also many attempts to evolve Wheeler's "It from bit" doctrine.

Regards,

Eugene.

Well, from Shannon's theorem to the plethora of nodes in the brain--all is information and nothing more!!

Our friend Bernd Schmeikal evolves the idea that space-time is informational structure. Now he is on the volcano island in Greece.

Well, I second the motion on the volcano island...there is simply no reason or rationale in this world, save living isolated and next to a volcano. 

From where is order then, Luisiana, and why does something exist at all?

Eugene, what a question! I prefer to resort to my innate knowledge given to me by some unknown Divine source that has only the intent of deceiving me... Think of the Cartesian circle and you'll get the blurb. 

Bernd successfully operates with these circles. He names them self-referencing nothing.

That moniker makes me chuckle... As the Tao de Ching says, "Better to hold fast to the Void."

What is nothing (void), Luisiana? It is the negation of all. It contains same amount of information as all.

Greetings from the volcano island!

About Climate Mechanics, which is the area that I study, it seems that the process is really chaotic, since all contemporary phenomenon is the result of consequences arising from past events that have created such conditions. It is very difficult to associate stochastic events in this case.

I disagree. The climate records show classic stationary and cyclostationary behavior, hence, are definitely stochastic in all regards. As for the question of chaotic...ultimately, no model will ever recreate the climate, much as weather models have failed miserably to recreate the weather; thus, back to the spirit of the question for this posting: "What practical difference lay between stochastic and chaotic systems?"

Hi Luisiana,

coming back to the discussion with a comment on your most recent answer, please accept my apologies for the lengthy (unintended) silence. What do you mean by "classic stationary and cyclostationary"? If you have an equilibrium system in mind, cyclostationarity certainly means the (otherwise stationary) system's linear response to a cyclic driver, notably (slow) astronomical forcing. How do you conceptually treat, however, free - not astronomically driven - oscillations of a feedback system at distance from equilibrium (or from stationary state)?

Take the issue of boreal summer monsoon prediction, for example. There is a substantial practical difference for farmers (and thus societies) living in the monsoon belt - and even for those in northern midlatitudes, at least - between an (unpredictable) stochastic change between drought and flood conditions and a system that exhibits organized dynamics (active-break monsoon cycle) based on synchronous motions of a hierarchy of planetary waves, as observed. Though monsoon prediction itself is notoriously difficult in general (mostly understood in terms of forecasting the system's total seasonal performance), it is not hopeless to intra- and inter-seasonally infer on the system's state a couple of weeks in advance - if one qualitatively understands its nonlinear (cyclic) behaviour and can grasp, say, its midsummer state of well-developed intraseasonal activity. Achievements in this respect would bear substantial agricultural relevance. And: though the system's autumn trajectory depends critically on the date where it passes the autumn monsoon bifurcation (according to my conceptual understanding), it can often be 'foreshadowed' even from a naive (and midlatitude) position without detailed knowledge of the tropic/subtropical processes. At a certain date late in August 2014, for example, it appeared clear that a long - say, Indian summer type - autumn may be expected. This was just a phase effect of monsoon dynamics, to my mind, but it produced one of the warmest years on record - only due to the length of the season, not due to especially strong heat wave(s) in summer.

Giving this example, I simply want to emphasize that conceptual understanding of emerging nonlinear dynamic structures is not just an 'academic' exercise, in general, without practical relevance.

Kind regards

Peter

Luisiana,

in the case of random numbers generators it is no less difficult to prove randomness, than periodicity. Actually, we can only estimate limitations on the period. So, your statements about randomness is not more definite, than somebody's about determenism. For example, if all is limited, then it can be periodically continued. Routine operation in the Fourier analysis.

Hold fast to the Void!

Hello Peter,

If you would insist upon chaotic descriptions for your forecasting in matters of the climate, then you would need to predict the monsoon season for the next decade! By your own admission...this is not possible; hence, to what method of forecasting do you rely upon, if chaotic modeling fails you in every regard?

Eugene,

That is exactly the difficulty regarding random sets...one can never prove beyond any doubt that a set is in fact random; but, one can show that a given set is indistinguishable from a random set, hence, mimics or reflects the behavior of a random set to such a degree that one may use that as a model for the set. The antithesis is also true...one may never prove determinism beyond all doubt, etc...

Once again, it is a matter of practicality, in that, if one is presented with a complex dynamic system, what tools are afforded to capture the characteristics of the system in an efficient and satisfactory manner. Of course, that is a loaded question, because much of what we call science is in fact the art of describing systems in some meaningful, efficient manner, regardless of whether or not the system's reality--as it is presented to us--is only apparent or onticalogically authentic.

You propose Turing's test for random sets, Luisiana? It is interesting, but as I know this test failed in the case of human machine difference.

It seems to me, that all these differences is the matter of practical convenience.

Eugene, convenience just might be the choice wording for most of scientific epistemology. In the end, one cannot know absolutely nor can anyone claim possession of that capital knowledge--Truth, with a capital T. 

In keeping with the question at hand: if a deterministic system can appear to be random, and vice versa, then I think it is more a question of convenience in all matters of modeling and prediction. 

Luisiana, your reply did hardly meet my reasoning. I gave you an example where it may be of practical relevance to understand emerging dynamic structures of a nonlinear climatic subsystem in detail and to exploit that knowledge for an intraseasonal 'foreshadowing' of a trajectory the system has chosen (be it a "chaotic" one or not). I did not adress the decadal timescale in this example. I simply said that a stochastic perspective is not too helpful in this agriculturally important task, and I mean this makes a "substantial and meaningful difference" between both views. Wasn't this just the question you posted? In this context, I also posed conceptually related questions concerning cyclostationarity etc. you did not touch as yet.

Sorry Peter...sorry for my neglect.

Take the pacific oscillation, for example, the ebb and flow of the hot and cold periods, the time it would take a mass of heated water to migrate geographically, the time for cold water to move and replace, the physical constraints lead one to easily think of this system as a deterministic dynamic system; thus, one can easily perceive this process to be a chaotic oscillator, but the periodicity, amplitude, etc, of the oscillations are highly sensitive, therefore, chaotic models would only suffice in academia and not any practical application. On the other hand, statistics would easily describe key characteristics and one would simply work within those parameters. If you imply that a chaotic model could be developed that accurately predicts such natural phenomena in advance of many years--I would disagree and experience bears this truth out. In other words, what use is a chaotic model, if you cannot predict the time dependent evolution of all sensitive parameters? For example, the Lorentz weather model, it is chaotic, deterministic, but all together impractical in actual use. Worse, the inherent jeopardy in placing faith in any such chaotic model, with made up parameter values for any given future state; furthermore, jeopardy exists not only in the made up factor, but in the high sensitivity these models have for parameter variation. If one resorts to ensemble averaging, then why waste time...just immediately employ stochastic models and be done with it.

Thanx for your prompt response, Luisiana. I would never claim that a chaotic system can be accurately predicted beyond a certain limit that is determined by the incomplete knowledge of an "initial state". You prefer a stochastic description since the system's dynamics cannot be forecast beyond such a limit.

Well, for that reason, as well as, for the following reason: one cannot even accurately determine the initial conditions! With chaotic systems, the sensitivity is so great, that a slight uncertainty in the initial conditions leads to a family of curves; therefore, do not trifle with such models! Stochastic models provide a global description, including all peripheral possibilities, but also the most likely outcome; thus, one has in hand a very useful set of predictors, indeed! 

... ctd, (sorry, i pressed the wrong button)

I would like to repeat, however, what also Lenny Smith pointed to: there are topological and geometric aspects of a nonlinear system's motion, and such more qualitative knowledge goes beyond what you may get from a stochastic description, and it may be of practical relevance to adopt this perspective. The monsoon's intraseasonal active-break cycle is a free oscillation of a planetary feedback system. It bears higher internal organization than you would expect from a stochastic system. And by the way: I'm not a friend of ensemble averaging .

... yes, a family of curves which for some period shadow each other. The qualitative dynamics of interest in my example is the change between seasonal drought and flood conditions, i.e. their timing. From a stochastic system you might get some probability density function which would hardly be helpful in "foreshadowing" the system's evolution a couple of weeks in advance.

Concerning ensemble averaging, I see the risk that interesting dynamic details are blurred - and lately I tend to agree that it may be wasting of resources. In my example, the spring-to-summer transition appears to bear the passage of a subcritical bifurcation, with a subsequent 'hard' transition into a chaotic July regime. In this situation, both the onset date and the initial phase angle of the established monsoon activity cycle are uncertain (perhaps a cause of the notorious "sping predictability barrier") and may differ from year to year in a model simulation (as it is the case in reality). Taking calendar averages of a set of such trajectories (i.e., a sort of ensemble average, maybe from a set of subsequent years) would 'smear' the characteristic oscillatory dynamics.

Peter, you elevate the discussion to a higher level of inquiry. Firstly, stochastic systems can and do enjoy a wealth of fine structure within their descriptions, such as self-similarity; albeit, the gross statistics of mean and variance are rather broad in their perspective. I disagree with regard to prediction and stochastic systems or, at least, their descriptions; because, if applied judiciously, one can--with great precision--predict weather events and other natural phenomena, like Earthquakes. In some sense, wisdom mimics this quite nicely, for isn't it the culmination of years of experience that enables the old farmer the ability to predict with uncanny accuracy the coming seasons and their severity. Similar "wisdom" is possible through judicious application of stochastic descriptors. 

I see you agree in all respects with "family of curves"; yet, who could pick the one curve that is right from a family of such curves? For any given chaotic description, one must admit a range of initial values, moreover, to all time dependent parameters another whole new set of possible values, hence, one is inevitably confronted with a range of possible outcomes. To which curve do I claim preeminence above and beyond any other and under what criteria? I cannot see any possible (rigorous) method for choosing any such one curve from a family of ergodic curves.

There are many such indications for astronomical influences for the climate, namely, Milankovitch cycles; nevertheless, it is with great trepidation that I would accept that any such "correlation" truly exists, furthermore, the degree of influence seems variable--as evidenced by the paleoclimate records--which leaves us, once again, quite poor for chaotic descriptors and, once again, we find ourselves ultimately resorting to that most reviled, hated and dreaded discipline--statistics. 

Hi Lenny, Luisiana,

I have no doubt that substantial stochastic aspects are to be considered when speaking about climate dynamics. Most interesting to me in this respect is the amplification in effect of small fluctuations, notably phase angle differences, due to the passage of bifurcation. Have a look, if you want, at the attached figure (Fig. 3.16 of "A General Circulation Model en route to intraseasonal monsoon chaos" among the featured papers of my RG account). The (model) system's autumn trajectories roughly fall into three categories of circulation, termed LN ("La Nina prone"), IS ("Indian summer type") and EN ("El Nino trigger") here, which are controlled in the case given by small differences in the circulation at the moment when the system passes the monsoon retreat bifurcation (displayed is the atmospheric mass anomaly over the northern hemisphere). Though the GCM's climate system plays a kind of roulette here, one may try to grasp its state shortly after passage of the bifurcation point (say, from day 270 on in the present example). "Foreshadowing" of the autumn trajectory after the fact (of passage of the bifurcation) is a reasonable scientific task then, with potentially substantial economic effects.

Conceptually, this also means that apparently strong stochastic effects in the dynamics of a nonlinear system may well be borne in topological (torus segment in phase space, in this example) and/or geometric properties (e.g., splitting of attracting manifolds, bifurcations) of the emerging phase space structures. And note please, Luisiana: I did not seek a chaotic description of monsoon dynamics: I found these structures in a circulation model that is based on the so-called primitive equations; that is, on the elementary transport equations of mass, momentum and energy (on a model Earth grid with very coarse spatial resolution). These nonlinear structures emerge from a primitive physical description and form a new level of description (understanding) - and even the notorious problem of successful monsoon simulation may well be borne in emerging structures of this type, including a full (inverse) 'route to chaos' that this model shows during the season.

As for the case of sensitive dependence on "initial conditions" of this model's autumn trajectory, note that the intraseasonal attractors in the back are not "chaotic" (strange) here. This points, by the way, to the fact that "sensitive dependence on initial conditions" is not a defining feature of chaotic systems: it is a symptom, not the very nature of chaos. But this is another theme ...

Let me conclude, for the moment, that deterministic and stochastic elements of a nonlinear system's behaviour are not that 'antagonistic' as it might appear ...

Best, Peter

Hi Peter

I am interested in your paper A General Circulation Model en route to intraseasonal monsoon chaos  Would you email it to my emialbox [email protected]. I studied East Asia summer monsoon onset in inerannual timescale. Thank you

I only want to point out the origin of chaos in weather was derived from using elementary, axiomatic models for mass transfer in the atmosphere; furthermore, it was also grudgingly admitted--after some time--the predictability of such a simple differential system was "unattainable". Moral: even though the system may be deterministic in principle, in practice, one may find no such modeling effort to avail.

Leonard, the ability to model out to a paltry four or five days hardly meets my personal criteria of "what more one could ask for?" In fact, the other perspective can be taken, that is to say, the chaotic models do not always accurately forecast, rather, there are a family of curves and in retrospection, one may find or "discover" that one of the curves comes 'close' to what actually happened. That is hardly forecasting! That is retrospectively congratulating oneself for hindsight bias. 

Hi Luisiana,

it appears that I did not yet fully grasp your point(s). Do you generally pose into question that advanced knowledge about emerging structure in a nonlinear dynamic system may have practical relevance, including improved capability to prediction? Do you mean there is no structure in chaotic dynamics of which improved knowledge may be of interest, so that one may replace its deterministic description with an unstructured stochastic one without loss of relevant information? Do you think, the nonlinear climate system ist either chaotic or in a laminar (i.e. largely linear) phase of its evolution, nothing inbetween? Do you mean the wisdom of old farmers, e.g., makes climate (etc.) science superfluous, be the system "classical stationary" or not? Wouldn't this lately mean science to be superfluous in general as compared with the wisdom collected over generations?

As far as I'm concerned, I am curious to learn about dynamics that generate "your" statistics, but I also learned to live in a scientific environment where my personal preferences do not count too much. Such is life ... by the way, a highly nonlinear, structured state of matter far apart from stationarity, of which the detailed nonlinear functioning may be of direct interest to many of us in certain periods and cases along the sometimes unpredictable (yet not necessarily uncontrollable) pathway, or trajectory, we follow ... :-)

Best, Peter

@ Leijang, you may download the final manuscript of the requested article from my RG account (cf. the featured articles there). This is the version that I'm allowed to store here and to distribute on a personal basis.

Peter,

 You address many facets of interest, but I think the statement, "...my personal preferences do not count too much.", addresses a very important quality of science; one that is rarely understood or recognized, that is, that science has largely become impersonal, as of late, and therefore enjoys total obscurity, from both public and scientists alike, for the majority of what is said goes unheard, unnoticed and, frankly, much of it is totally unnecessary. 

Luisiana,

I should have been a bit more specific in this sentence, though it was not meant too seriously. What I have experienced is, say, that the "statistics community" dominated the climate science for (too) long, and dynamic systems approaches (also your target) remained largely unheared, except in smaller circles. Among the effects of such ignorance was the following: Until 2001 at least, the IPCC reports were dominated by the (northern) midlatitude perspective, and one literally had problems to even find the term "monsoon" therein - a phenomenon that directly concerns billions of people. The large GCMs did not properly simulate the monsoons, but statistical measures were found then to show that all is in reasonable order. Nothing was in order, the GCMs were (are?) apparently run in  inappropriate operating regimes. Another, certainly related effect may be observed at present: None of the GCMs used for climate projection appears to have shown even the weakest hint at the thermal stagnation (the "hiatus").since the 1990s. Now, the multidecadal oscillations are being "observed". Before, the search was for ever better forcings that could make the models match the climate history of the instrumental period. Linear thinking, in effect, no serious acknowledgement of a sort of regime character of climate dynamics - which can be seen if one looks at the data with the naked eye. Ignoring the nonlinear character of climate dynamics is an irresponsible  behaviour of scientists, to say it frankly as well.

All right, so, at one point in time, climate was primarily described in statistical terms, then, around the 1980s, a major thrust was made to develop chaotic descriptors, in the hope of finding a better predictor. Now, placing aside the all-too-often misapplication of statistics, where scientist fail to properly understand the principles and theory, hence, misapply these axioms in a manner not sufficient for descriptions; setting all that aside, the IPCC and their GCM models have failed miserably to predict anything of any worth, they have in fact, failed absolutely! These models are wide and varied, but they are largely deterministic-based models, with a set of dose-response functions, integrated over space and time, etc... The fact that these models failed to incorporate the Monsoon effect is a tell-tale sign these models hold no validity to the actual climate, they are simply heuristic in nature, irrespective of their apparent 'exactness' of deterministic descriptors. Being heuristic in nature simply means that these models will be coaxed into some sort of 'agreement' with actual data; but, the climate will wander randomly over time and the modelers are forever going to 'update' their bloody models.

One thing regarding nonlinearity: there has been a real obsession with nonlinear descriptions, but these models are usually intractable, fail to provide a faithful global solution and their utility is confined to very narrow, overspecialized set of circumstances, which would per chance rarely occur in any frequent manner. 

For me, nonlinearity is failure to properly identify the underlying linear relationships and properly capitalize upon those solvable problems.

Setting all aside which is nonlinear and difficult to treat, and what either statistians or modellers who prefer deterministic (primitive or conceptual) modelling failed to properly describe in their terms, what remains are linear, statistically solvable problems. Bravo! If I lost a key in the dark, I will seek for it in the light circle of the street lamp - and probably come to the conclusion that I did not loose a key at all. Luisiana, have a look around to see that the environment we live in is full of structures that emerge from elementary rules of local interaction but can hardly be expected to exist at this level of description. If one wants to label it this way, it appears to be the "unreasonable effectiveness of mathematics in physical sciences" (Wheeler), which bears this higher level of structuring and description - among the mathematical disciplines notably number theory, which is 'unreasonably effective' in describung synchronization phenomena (low rational frequency relationships, for example, at top of the Farey tree of rational numbers). Cf. the paper of Lagarias ("number theory and dynamical systems") which I like to cite in this context.

Concerning GCMs, they contain all the ingedients needed, but - you're certainly right in this respect - may become intractable (or at least difficult to handle) when chaotic dynamics, homoclinic or heteroclinic orbits, etc. emerge. The "solution" of fixing the srews that parameterizations offer until dynamics become tractable just leads to the problem that I have quoted (as an example) of inadequate monsoon simulation. I can demonstrate this with the "small" GCM that I have run in the past in order to better understand these dynamics (a Mintz-Arakawa GCM). I have learned a lot when using this model, even of its failures galore (instabilities) I had to struggle with. I could not have developed my conceptual view on boreal summer monsoon dynamics without experimenting with that 'old-fashioned' model. So, at least for me, it was very helpful to have such a tool near at hand (which could be run even on a 386 PC at the beginning of the 1990s).

A nonlinear system of equations bears the corresponding linear solutions as well, and if the situation (initial and boundary conditions, parameter values) is such, it will find them - but reducing it to linearity in cancelling the nonlinear terms, you will never see the nonlinear solutions, of course, and thus perhaps come to the conclusion ... the key has not been lost :-) (my apologies). The problems I quoted of present-day large GCMs appear (to me) to be a result of "fixing the screws" until the models become tractable (an economic decision as well), and losing the interesting nonlinear dynamics this way that at least qualitatively correspond to the behaviour of mother nature.

Kind regards,

Peter

Peter is right, Luisiana. It is evident, that linearity and statistics are not enough.

Regards,

Eugene.

I recognize that the world is far more complex than a linear graph; but, in the same spirit as Kant, I claim Man cannot understand anything more complex than a two-dimensional graph. Even better than that, Kant once said, "You do not truly understand, until you can explain it to your grandmother." 

Nonlinearity is usually bound by an envelope; that envelope can be approximated by suitable linear means. If any claim to be able to derive a multi-valued function describing N-body particles freely moving about...well, in short, it is not possible; regardless of any desire found nestled within the heart of Man. 

My 3D capabilities are limited indeed, but I recognize that there are many people, especially younger ones, who are able to solve the problem posed by Rubik's cube, for example - a hopeless task for me personally. But I can imagine the motion of a system on the mantle surface of a torus, and the 2D projections of it - and having this "talent" ... which also my grandmother (the one to whom I remember) certainly had if she was a bicycle driver (which I don't know, unfortunately, since she passed away already in the 1950s) ... I can also imagine to some degree what happens in 3D, as well as in these projections, if the torus becomes folded, streched, bumby etc. There exists a simple geometrical understanding of a Hopf bifurcation, be it a super- or subcritical one. And so on ... These are the ingredients I needed to develop a topological / geometrical, i.e. qualitative, understanding of the behaviour of the planetary monsoon system in boreal summer, if displayed in terms of the climate system's integrals of motion (an appropriate level of description if the very nature of the system is of interest).

I recommend the book "Dynamics. The Geometry of behaviour" by Abraham & Shaw, which consists of a large collection of graphs which show solutions of many nonlinear problems, without any further mathematics - just to help develop a geometrical understanding ... that goes beyond the claims you cite, Luisiana, of Kant (which I'm not sure he would maintain if living today, however). And also note please: a complex dissipative feedback system may behave low(er)-dimensionally; that is, may not exploit its formal degrees of freedom in full. Laplace is not the best reference and authority in case of feedbacks and synchronous motions - Huygens would be a better one.

All good points, indeed; but, we are moving far afield of the question at hand. Mind you, any discussion can widen in its scope and one can mention certain illustrious personages along the way; but, this does nothing in solving a problem. 

Riemann surfaces are particularly interesting in their scope; but, topologically speaking, one can only cover a simply connected space--adequately. If one spans many branches, then we are swiftly getting into that space that causes so much consternation, both in representation and mastery. 

I'll switch my approach: given the existing GCM models, which purports to model the climate in all its intricacy, then what would explain their failure in adequately representing the climate evolution? In other words, their failure is an indication of their inadequacy.

Is a 'failure' just an invitation for further complication...or does it signify that "one is trying to grasp far too much?"  

Hello Leonard,

Interesting points... In response, I'll say, "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy" It is fallacious to think that Man has deciphered Nature thoroughly. 

OK. So, as with everything, just stating something has very little meaning. So, I spent the time to look up some paper on the accuracy for various planetary models, see attached. Typical of models, whether numerical or otherwise, where the accuracy and precision is not as good as one might tout. Keep in mind, these models, model a rather simple system of huge planets; hence, one should think that we could model their movements with the greatest of accuracy. The models show large deviations for a simple period of time stretching some 50 years. 

Now, consider the climate, with so much of it not understood, plus, so much of it beyond any reasonable capture; thus, I'll state the obvious, "The GCM models have failed miserably." This despite all the noise regarding Hopf bifurcation, chaotic dynamics and a host of things as of yet mentioned.

Can you explain to me, please, the elementary logic behind your obvious claim that the failure of GCMs backs your noise about nonlinearity etc, given the fact that alternative reasoning may be found in my noise, for example?

Slow down! Everyone is approaching the question from their own point of view...and there is no "right" or "wrong"--done. 

Let me speak of a more recent failure of climate science in its entirety: scientists have recently declared the onset of El Nino. Now, they had to wait for the onset before they could claim El Nino is upon us; hence, their models could not predict the onset of El Nino. With that said, there is a litany of predictions released upon the news that El Nino is upon us. Of course, these 'predictions' were generated by using as initial conditions the present set of circumstances. OK. Not very impressive. One must wait for the natural occurrence to occur, before one can 'accurately' 'predict' the occurrence of said natural phenomena. Worse, this is a major climatic event, the El Nino cycle, yet no climate model can accurately predict the onset, let alone the duration, intensity, etc...

If these points are not indicative of a model that is a failure in all respects, then what is?

point of reference: http://www.japantimes.co.jp/life/2015/05/13/environment/aussie-scientists-declare-el-nino-onset-forecast-substantial-weather-extremes/#.VWJqo4708UR

here's another: I mean, I'm not scouring the internet to find these!!

http://search.proquest.com/openview/6f7d6dc22f47559806b4a4ee15ee9341/1?pq-origsite=gscholar

And this one highlights the 'frustration' climate scientists feel: http://www.slate.com/blogs/future_tense/2014/11/06/el_ni_o_prediction_2014_why_weather_forecasters_were_wrong_about_a_super.html

This paper explores ways of trying to improve GCMs, but admits their performance no better than statistical models: https://www.researchgate.net/profile/Lisa_Goddard/publication/253013751_Seeking_Progress_in_El_Nio_Prediction/links/02e7e53ba9e857c15a000000.pdf

Article Seeking Progress in El Niño Prediction

Let me prime the pump by approaching the issue from another angle. Consider a nonlinear oscillator, with some tight range of frequencies. Now, if observations prove the variance of the actual occurrences for the peaks (the phase) either equals or exceeds the range of frequencies fundamental to the system, then pray tell, how could you ever prove the system to be governed by said nonlinear chaotic equation?

Have You seen this, Luisiana?

All right. Attached is a really good paper of the issue of stochasticity and chaotic systems...enjoy!

Eugene, we've all seen the occurrence of an oscillator with random phase change. The thing is--just because you can 'fit' a chaotic curve to the data, does not necessarily mean that you have identified the underlying physics. One can use any basis set under the Sun and fit or spline fit that to any data set. Fitting does not mean equal to or identification.

Also, I have seen this with the onset of El Nino, which exhibits varying time periods, who's variance is about equal to the duration of the occurrence, hence, it is very difficult to prove that El Nino's are in fact chaotic in their occurrence or behavior; contrary, it is quite the opposite, one can more easily prove that these cycles are random in nature, with no known indicators, like temperature, etc... 

I understand You, Luisiana, but the question "From where is order?" remains.

We can prove, for example, that there can not be any chaos in two dimensions.

I was considering processes occurring within the climate.

I present the following hypothesis for "whence does order come?" 

"By virtue of system constraints, does order arise and is nothing more."