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I'm trying to figure out what are the theoretical and practical, implications and limitations, when a high-dimensional chaotic process is modeled as a random process. I understand how low-dimensional chaos (callin' it LD chaos from now on) is different from a stochastic process. However I'm unclear on the pitfalls of approximating high-dimensional chaos (HD chaos) as a random process, in terms of linear stochastic PDEs.

In my field of earth system modeling, weather systems or ocean mesoscale turbulence are (i think) good examples of HD chaos. Some modelers will use stochastic processes to represent unresolved turbulence. For example, a noisy diffusion equation has been studied to describe the statistical evolution of the ocean surface temperature field $T(x,t)$ $$\frac{dT(x,t)}{dt}=\kappa\nabla^2 T(x,t)+\eta(x,t)$$ $$<\eta>=0$$ $$<\eta(x,y)\eta(x',t')>\propto \delta(x-x')\delta(t-t')$$ where $\eta$ is a white noise representing turbulent weather forcing from the atmosphere. Or likewise you could have a conservative noise term appearing in the temperature flux.

If the process that $\eta$ represents is really LD chaos, I see how this would be a bad model as LD chaotic dynamics are governed by an attractor, which is definitely not random.

My question then is, if $\eta$ is a stochastic process that is approximating HD chaos, is this approximation going to lead to trouble or caveats? I know you could also do things like change the autocorrelation function (maybe to decay as a power-law), but is that actually enough? Is HD chaos actually distinguishable from noise? Does HD chaos have some complicated attractor? If HD chaos is represented as a random process, can it be proven that this affects the statistics of the solution?

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TL;DR: Every sufficiently complex deterministicity is indistinguishable from stochasticity.

Randomness in modelling is almost always used as an approximation of an intricate process that is too complicated or tedious for you to model or about which you do not have enough information. (The main exception is when we go down to the quantum level, where deterministicity may not be given.)

Also consider the following:

  • There are simple chaotic systems (e.g., the Zaslavsky map) that generate time series that approximate the fundamental properties of white uniform noise (uniform amplitude distribution, instantly decaying autocorrelation function).

  • Every pseudo random-number generator is nothing but a complicated chaotic map, if you so wish. (Strictly speaking, it’s not even chaotic but periodic with a very high period length; also it only has discrete states. However, this also applies to any digital realisation of a chaotic process.)

  • Every deterministic type of model is just a special case of a stochastic model. SDEs contain ODEs as a special case; Markov chains contain simple maps as a special case; and so on.

To address your individual questions:

  • My question then is, if $η$ is a stochastic process that is approximating HD chaos, is this approximation going to lead to trouble or caveats?

    Of course, if you can replace your stochastic process by a more intricate model of what is actually going on, this gives you a better approximation of reality, but beyond that I am not aware of any problem caused by using an appropriate stochastic process.

  • Is HD chaos actually distinguishable from noise?

    If you dig deep enough and know where to look: sure. But if the noise is chosen appropriately, this should not affect the quality of your model.

  • Does HD chaos have some complicated attractor?

    Unless it’s transient chaos, yes. But why would this affect your situation?

  • If HD chaos is represented as a random process, can it be proven that this affects the statistics of the solution?

    The entire point of a stochastic model is to reflect the statistical properties of some real-world process. If relevant statistical properties are affected by using a random process, the latter is not chosen well enough. As already mentioned, if you dig deep enough, you can always find some statistical property that is crucially different.

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    $\begingroup$ I get your point so far. I found a paper i had been looking for (but only quasi understand). The paper is "A Fundamental Limitation of Markov Models" [DelSole., T, 2000] in J. Atmo. Sci. They show correlation functions for deterministic dynamic and Markov models. At "short times", Markov models correlation differs/fails to model covariance in the deterministic system. Has it been shown (besides this paper), that misrepresenting the correlation at small time lags has dynamical consequences in a numerical model? Can small scale inaccuracies limit large scale (possibly even emergent) behavior? $\endgroup$ – Z W Jul 24 '17 at 20:07
  • $\begingroup$ @beepboop: At "short times", Markov models correlation differs/fails to model covariance […] – I skimmed the article and as far as I understand it, the problems of Markov models arise from being discrete in time and having no memory (in fact, the process governed by any ODE is a special case of a continuous-time Markov process; also see my edit). — Has it been shown (besides this paper), that misrepresenting the correlation at small time lags has dynamical consequences in a numerical model? – I am not sufficiently familiar with stochastic processes to answer that question. $\endgroup$ – Wrzlprmft Jul 25 '17 at 9:44

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