# Lyapunov Exponents, Smooth Conjugacy, and Chaotic Behavior for Random Dynamical Systems

Wednesday, September 19, 2012 - 9:00am - 10:30am

Keller 3-180

Kening Lu (Brigham Young University)

In the first lecture, we study the Lyapunov exponents and their associated invariant

subspaces for random dynamical systems, which are generated by, for example, stochastic or random differential equations. We present a multiplicative ergodic theorem for infinite dimensional random dynamical systems..

In the second lecture, we present several theorems on smooth conjugacy for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates which are used in the construction of conjugacy.

In the third lecture, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We show that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.

subspaces for random dynamical systems, which are generated by, for example, stochastic or random differential equations. We present a multiplicative ergodic theorem for infinite dimensional random dynamical systems..

In the second lecture, we present several theorems on smooth conjugacy for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates which are used in the construction of conjugacy.

In the third lecture, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We show that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.

MSC Code:

34D08

Keywords: