1 Chaos in Differential Equations Driven By Brownian Motions Qiudong Wang, University of Arizona (Joint With Kening Lu, BYU)

2 1. Equation Driven by Brownian Motion  Random Force by Brownian Motion.  Wiener probability space Open compact topology  Wiener shift:  Brownian motion r

3 1. Equation Driven By Browian Motion  Random forcing r

4 1. Equation driven by Brwonian Motion  Unforced Equations:

5 1. Equation Driven by Brwonian Motion  Assume

6 1. Equation Driven by Brwonian Motion  Equation Driven by Random Force: where Multiplicative noise, singular, unbounded.

7  Random Poincare Return Maps in Extended Space Poincare Return Map 2. Statement of Results

8 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 Results

9 Corollary A. (Duffing equation) the randomly forced Duffing equation has a topological horseshoe of infinitely many branches almost surely.

10 2. Staement of Results Corollary B. (Pendulum equation) the randomly forced pendulum equation has a topological horseshoe of infinitely many branches almost surely.

11 3. Idea 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.

12  Topological Horseshoe:

13 (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.

14 (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)