Generalized Hamiltonian norm, Lyapunov exponent and stochastic stability for quasi-Hamiltonian systems

Abstract

The generalized Hamiltonian norm and the corresponding largest Lyapunov exponent are defined in this Letter. The largest Lyapunov exponent in terms of the generalized Hamiltonian norm can take into account system nonlinearity and approaches the largest Lyapunov exponent in terms of the Euclidean norm with system states tending to zero in the case of non-degenerated Hamiltonian. By using the stochastic averaging method, the largest Lyapunov exponent can be evaluated approximately for stochastic stability analysis. If the non-conservative forces are expressed as functions of Hamiltonian or integrals of motion, the largest Lyapunov exponent of the averaged system is equal to that of the original system under the compatibility conditions. A quasi-integrable Hamiltonian system subjected to randomly parametric excitations is analyzed to illustrate the generalized Hamiltonian norm and the corresponding largest Lyapunov exponent for stochastic asymptotic stability.