Reliability of quasi integrable generalized Hamiltonian systems

The reliability of quasi-integrable generalized Hamiltonian systems is studied. An mm-dimensional integrable generalized Hamiltonian system has MM Casimir functions C1,…,CMC1,…,CM and nn (n=(m−M)/2n=(m−M)/2) independent first integrals HM+1,…,HM+nHM+1,…,HM+n in involution. When an integrable generalized Hamiltonian system is subjected to light dampings and weakly stochastic excitations, it becomes a quasi-integrable generalized Hamiltonian system. The averaged Itô equations for slowly processes C1,…,CM,HM+1,…,HM+nC1,…,CM,HM+1,…,HM+n can be obtained by using stochastic averaging method, from which a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established. The conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, an example of a 5-dimensional quasi-integrable generalized Hamiltonian system is worked out in detail and the solutions are confirmed by using a Monte Carlo simulation of the original system.