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Applied
Math Seminar
Spring Quarter 2002
3:15 - 4:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C
Friday, April 5, 2002
Peter H. Baxendale
(University of Southern California)
Lyapunov exponents for small random
perturbations of Hamiltonian systems
Abstract:
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Consider the stochastic nonlinear oscillator equation
$$
\ddot{x} = -x -x^3 + \varepsilon^2 \beta \dot{x} + \varepsilon \sigma
x \dot{W}_t
$$
with $\beta < 0$ and $\sigma \neq 0$. An old result of Auslender and Milstein
shows that the top Lyapunov exponent $\lambda(\varepsilon)$ for the system
linearized at 0 satisfies $$
\lambda(\varepsilon) = \varepsilon^2 \frac{(4 \beta + \sigma^2)}{8} +
O(\varepsilon^4) \mbox{ as }\varepsilon \to 0
$$
If $4\beta + \sigma^2 > 0$ then for small enough $\varepsilon > 0$ the
system $(x,\dot{x})$ is positive recurrent in ${\bf R}^2 \setminus \{(0,0)\}$.
Now let $\overline{\lambda}(\varepsilon)$ denote the top Lyapunov exponent
for the linearization of this equation along these recurrent trajectories
in ${\bf R}^2 \setminus \{(0,0)\}$. Then $$
\overline{\lambda}(\varepsilon) = \varepsilon^{2/3}\overline{\lambda}
+ O(\varepsilon^{4/3}) \mbox{ as }\varepsilon \to 0
$$
with $\overline{\lambda} > 0$. This recent result is joint work with Levon
Goukasian (USC). The appearance of the exponent $2/3$ depends crucially
on the fact that the system above is a small perturbation of a Hamiltonian
system. The method of proof can be applied to a more general class of
small perturbations of two-dimensional Hamiltonian systems. The techniques
used include an extension of results of Pinsky and Wihstutz for perturbations
of nilpotent linear systems, and a stochastic averaging argument involving
motions on three different time scales.
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