# A Are Chaotic and Stochastic processes related?

#### JorgeM

Hello everyone.
I have read on the web some people that mention something about "stochastic chaos" but I am not that sure about what it really means or if that actually exists. Two months ago , I started to study some chaotic systems but in stochastic systems I am not that familiarized in order to figure out how may be this two math areas related.
If someone could tell me ( or to explain) if both are related somehow or if that is actually impossible, I would be grateful.
Thanks for reading my post.
Mtze

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#### S.G. Janssens

Science Advisor
Education Advisor
There is a lot to say about this, but this has been done better in the literature, so I will be brief.

A purely deterministic dynamical system - for example, a map from a complete metric space to itself, or a (semi)flow defined on a metric space - can display chaotic behavior in a well-defined sense.

One way to understand this behavior, is by studying the associated Frobenius-Perron and Koopman operators (physicists sometimes speak of "transfer operators") or more generally Markov operators that act on a space of (usually absolutely continuous) measures. For this perspective, the book by Lasota and Mackey, Chaos, Fractals and Noise is highly recommended as an introduction to "applied" ergodic theory. (Do not let the catchy title fool you.)

Of course, it is possible to start with a deterministic system and introduce a co-dependence on some discrete or continuous time stochastic process to obtain a stochastic dynamical system. The associated operator on a space of measures is called the Foias operator.

My recommendation would be to first understand stochasticity of purely deterministic dynamical systems by studying the associated measure or density dynamics along the above lines, and only then move forward to include stochasticity at the level of the original deterministic dynamical system.

#### JorgeM

Thanks for your reply, I will try to start with the purely deterministic one, and I am gonna read the books you menctioned. Thanks a lot for your advise!

#### WWGD

Science Advisor
Gold Member
As I understand it a chaotic system is a dynamical system in which the limit state is fractal, i.e., of non-integer Hausdorff dimension. A Stochastic process is just a random process, i.e., described by a Random variable.

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