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Qiudong Wang, University of Arizona (Joint With Kening Lu, BYU). Chaos in Differential E quations Driven By Brownian Motions. 1 . Equation Driven by Brownian Motion. Random Force by Brownian Motion. Wiener probability space Open compact topology Wiener shift: Brownian motion r. - PowerPoint PPT Presentation
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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)