Instability of the Harmonic Oscillator with Small Noise

We construct asymptotic expansions for the exponential growth rate (Lyapunov exponent) and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous Markov process with a finite number of states. In the case of two states (the telegraph process) we obtain additional terms in the expansions, affording comparison between the exact values and the asymptotic formulas.

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