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Chaos in Chaos in Differential Differential EEquations Driven By quations Driven By Brownian Motions Brownian Motions

Qiudong Wang, University of ArizonaQiudong Wang, University of Arizona (Joint With Kening Lu, BYU) (Joint With Kening Lu, BYU)

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11. . Equation Driven by Brownian MotionEquation Driven by Brownian Motion

Random Force by Brownian Motion.

Wiener probability space

Open compact topology

Wiener shift:

Brownian motion

r

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1. Equation Driven By Browian Motion1. Equation Driven By Browian Motion

Random forcing

r

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1. Equation driven by Brwonian Motion1. Equation driven by Brwonian Motion

Unforced Equations:

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1. Equation Driven by Brwonian Motion1. Equation Driven by Brwonian Motion

Assume

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1. Equation Driven by Brwonian Motion1. Equation Driven by Brwonian Motion

Equation Driven by Random Force:

where

Multiplicative noise, singular, unbounded.

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Random Poincare Return Maps in Extended Space

Poincare Return Map

22. Statement of . Statement of ResultsResults

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Theorem. (Chaos almost surely)

has a topological horseshoe of infinitely many branches almost surely.

• Sensitive dependence on initial time.

• Sensitive dependence on initial position

2. Statement of Results2. Statement of Results

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2. Statement of Results2. Statement of Results

Corollary A. (Duffing equation)

the randomly forced Duffing equation

has a topological horseshoe of infinitely many branches

almost surely.

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2. Staement of Results2. Staement of Results

Corollary B. (Pendulum equation)

the randomly forced pendulum equation

has a topological horseshoe of infinitely many branches

almost surely.

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33. . Idea of ProofIdea of Proof

(A) How to describe the chaotic dynamics for non-autonomous equation

without any time-periodicity?

--- The Poincare return map defined on an infinite 2D

strip in the extended phase space.

--- Obtain an extension of Smale’s horseshoe using

vertical and horizontal strips.

--- A Melnikov-like method for non-autonomous

equations without any period in time.

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Topological Horseshoe:

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(B) Brownian motion is unbounded. They can not be treated as

perturbations!

--- Usual dynamical structure, such as stable and unstable

manifold, Melnikov method, are all based on theory of

perturbations.

--- Instead of stable and unstable manifolds, we only have

stable and unstable fragments.

--- We need to find, and match these fragment to create

horseshoe.

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(C) How to prove the existence of chaos for ALMOST ALL paths

with respect to the Wiener measure?

--- Ergodicity of the Wiener shift is critical.

--- Need to compute the expectation and the variance of the

random Melnikov function over all sample paths in Wiener Space.

--- Finally, need a recent local linearization results proved by

Kening Lu for stochastic equations.

A theory on non-autonomouos equations

We study the equation in the form of

The theory of rank one attractors I have developed with Lai-Sang Young in last ten years apply.

(joint with W. Ott)

A way similar to Melnikov’s method to verify the existence of strange attractors dominated by sinks, strange attractors with SRB measures in given equations. Theory went far beyond Smale’s horseshoe.

(joint with Ali Oksasoglu)