Lyapunov Exponents and Stochastic Stability of Quasi-Integrable-Hamiltonian Systems

Abstract

The averaged equations of integrable and nonresonant Hamiltonian systems of multi-degree-of-freedom subject to light damping and real noise excitations of small intensities are first derived. Then, the expression for the largest Lyapunov exponent of the square root of the Hamiltonian is formulated by generalizing the well-known procedure due to Khasminskii to the averaged equations, from which the stochastic stability and bifurcation phenomena of the original systems can be determined approximately. Linear and nonlinear stochastic systems of two degrees-of-freedom are investigated to illustrate the application of the proposed combination approach of the stochastic averaging method for quasi-integrable Hamiltonian systems and Khasminskii’s procedure.