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Assume I have a dynamical system with additive process noise of the form

$$\mathbf{x}_{t} = \mathbf{F}\left(\mathbf{x_{t-1}}\right) + \mathbf{\epsilon}$$

where $\mathbf{x}_{t}$ is the state at time $t$ and $\mathbf{\epsilon}$ is some form of additive white/uncorrelated noise. I know that if the forecast $\mathbf{F}$ is dissipative (for example an autoregressive process), states $\mathbf{x}_{s}$ and $\mathbf{x}_{t}$ will decorrelate increasingly with $t$ and $s$ far apart, due to the influence of the additive noise.

What I am not sure about is whether the same rule holds for chaotic systems. Is this the case if $\mathbf{F}$ is chaotic as well?

J.Galt
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    I suspect that depends on what you can say about the phase space and the distribution of initial conditions. For example, the existence of an attractor may lead to highly correlated futures over a measurable subspace of initial conditions. The properties of the noise should matter as well. – DifferentialPleiometry Oct 30 '21 at 01:40
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    Maybe try this idea out on the [Lorenz](https://en.wikipedia.org/wiki/Lorenz_system) with whatever noise model you are interested in. – DifferentialPleiometry Oct 30 '21 at 01:44
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    So, suppose $F=1$, then you could day it doesn't decorrelate, because the memory is infinite. Is this what you're thinking of? – Aksakal Oct 30 '21 at 02:12
  • @Aksakal: It depends. If I start with, say, samples from a correlated prior, then repeatedly add uncorrelated noise, I would expect the state distribution's correlation to weaken with time. I have clarified that I mean white noise in the question above (a good comment from Galen!) – J.Galt Oct 30 '21 at 02:45
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    What do yo mean by chaotic? The temperatures in Germany and France behave chaoticly, but are still correlated for future times (because the rotation of the earth around the sun, which is part of the state, is not so chaotic). – Sextus Empiricus Oct 30 '21 at 09:30
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    In the example I gave the impact of x(s) on future state x(t+s) doesn’t change with time t passed if measured as d x(t+s) / d x(s), it is always equal to 1. So I’d say it doesn’t decorrelate in this sense – Aksakal Oct 30 '21 at 17:23
  • @Sextus Empiricus: Ah, yes, perhaps I should clarify. In my example, the variables/dimensions I consider correlation over would be the states $\mathbf{x}_{t}$ at different times $t$, so (for example) seasonal effects would mostly be an irrelevant mean shift. Imagine rather that I approximate the chaotic dynamics with Monte Carlo samples: in that case, the measure of "decorrelation" would be how much the sample trajectories have been "scrambled" between the two time steps, e.g., that samples which were previously in the middle of the distribution may now be at its edge, and so forth. – J.Galt Oct 31 '21 at 12:44
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    @Aksakal: That's a fair point, thank you! I'm starting to fathom how non-trivial this question I've posed ends up being ^^ – J.Galt Oct 31 '21 at 12:47

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