EXACT STATIONARY SOLUTIONS OF STOCHASTICALLY EXCITED AND DISSIPATED INTEGRABLE HAMILTONIAN-SYSTEMS

It is shown that the structure and property of the exact stationary so lution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. F or a stochastically excited and dissipated nonintegrable Hamiltonian s ystem, the exact stationary solution is a functional of the Hamiltonia n and has the property of equipartition of energy. For a stochasticall y excited and dissipated integrable Hamiltonian system, the exact stat ionary solution is a functional of n independent integrals of motion o r n action variables of the modified Zhejiang University, Hamiltonian system in nonresonant case, or a functional of both n action variables and alpha combinations of phase angles in resonant case with alpha (1 less than or equal to alpha less than or equal to n - 1) resonant rel ations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solu tions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian system s, which are further generalized to account for the modification of th e Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the ex act stationary solutions of stochastically excited and dissipated inte grable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. T he above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissip ated Hamiltonian systems as special cases, Examples are given to illus trate the applications of the procedures.