Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

Abstract

It is shown that the structure and property of the exact stationary solution 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. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ⩽ n – 1) resonant relations, 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 solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

Exact Stationary Solutions of Stochastically and Harmonically Excited and Dissipated Integrable Hamiltonian Systems

“…(iii) Exact stationary probability density (13) is usually almost periodic functions of time t, while stationary probability density (18) may be periodic and may be time independent when the associated Hamiltonian system is completely resonant. (iv) Exact stationary solutions (13) and (18) are reduced to those in reference [11] in absence of deterministic excitation g G "0. …”

“…Taking account of non-negativeness of p and the boundary conditions in Eq. (12), the exact stationary solution to FPK equation (11) in this case is assumed to be of the form…”

“…non-linear stochastic systems, it is advantageous to formulate the systems as stochastically excited and dissipated Hamiltonian systems [7}10]. Recently, it was shown by Zhu and Yang [11] and Zhu and Huang [12] that the functional form of the exact stationary solution of an n d.o.f. stochastically excited and dissipated Hamiltonian system depends upon the number of integrals of motion in involution and the number of resonant relations in the associated Hamiltonian system.…”

“…The exact stationary solution for the stochastically and harmonically excited integrable Hamiltonian system (10) is the exact solution of FPK equation (11) together with boundary conditions in equation (12) and the periodic conditions with respect to G when tPR. The di!erence between the transient and stationary solutions of FPK equation (11) in this case is that the former must also satisfy the initial condition while the latter does not have to satisfy the initial condition. In the following the functional form of the exact stationary solution for stochastically and harmonically excited integrable Hamiltonian system (10) with non-resonant or resonant associated Hamiltonian system and the procedures to obtain them are developed.…”

Some recent advances in the theory of random vibration

“…Additional contributions can be found, for example, in the works of Zhu and associates (e.g. Zhu and Yang 1996) and Soize (1994).…”