Lyapunov Exponents and Rotation Numbers of Linear Systems with Real Noise
Abstract
In the joint work with Volker of the University of North Carolina, we have investigated the Lyapunov stability of systems defined by a system of differential equations with a stochastic driving term, which may be either white noise or real noise. In the first case we showed that for nilpotent systems it is possible to compute an arbitrary number of terms in the asymptotic expansion of Lyapunov exponent in fractional powers of the noise coefficient when this tends to zero. This includes the important case of critically damped oscillator, which had not been treated previously. These results were then extended to the case of the same nilpotent system driven by a finite-state Markov noise process. This was obtained by a method of homogenization, using techniques previously established to study the central limit theorem for functions of a centered Markov chain. It is shown, as in the case of white noise, that the Lyapunov exponent admits an expansion in fractional powers of the noise parameter, and that the first term of this expansion agrees exactly with the result obtain in the white noise case.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 31, 1990
- Accession Number
- ADA234729
Entities
People
- Mark Pinsky
- Volker Wihstutz
Organizations
- Northwestern University