MAXIMAL LYAPUNOV EXPONENT AND ROTATION NUMBERS FOR 2 COUPLED OSCILLATORS DRIVEN BY REAL NOISE

Asymptotic expansions for the exponential growth rate, known as the Ly apunov exponent, and rotation numbers for two coupled oscillators driv en by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dy namical systems and in parametrically excited linear mechanical system s. Almost-sure stability or instability of dynamical systems depends o n the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator asso ciated with real noise provided that the natural frequencies are nonco mmensurable. The results presented here for the case of the infinitesi mal generator having a simple zero eigenvalue agree with recent result s obtained by stochastic averaging, where approximate Ito equations in amplitudes and phases are obtained in the sense of weak convergence.