On the Lyapunov Exponent of a Harmonic Oscillator Driven by a Finite-State Markov Process
Arie Leizarowitz · SIAM Journal on Applied Mathematics · 1989
This paper considers the asymptotic behavior of the random harmonic oscillator driven by a Markov process with a finite-state space. For a certain choice of variables, the motion of the system follows circles in the plane, with jumps from one circle to another at random times. The Lyapunov exponent is expressed in terms of the equilibrium measure of a related Markov chain. The latter satisfies an integralequation whose form permits an easy derivation of estimates in various limiting cases. This is demonstrated by computing the first-order approximation in a limit of weak noise. It is shown how some of the results can be extended to a certain class of two-dimensional systems.