A symplectic homotopy perturbation method for stochastic and interval Hamiltonian systems and its applications in structural dynamic systems

This paper presents a symplectic homotopy perturbation method for Hamiltonian systems with uncertainties. The effects of uncertainties cannot be ignored, but the research on symplectic algorithms of Hamiltonian systems with uncertainties is still weak. Besides, the nonlinear dynamic systems with uncertainties are far more complicated than deterministic linear systems. In this paper, taking two kinds of uncertainties into consideration, the dynamic responses of stochastic and interval Hamiltonian systems, especially uncertain nonlinear Hamiltonian systems, are investigated based on the homotopy perturbation method, respectively. By introducing an embedding parameter, a series of homotopy perturbation equations are deduced, and the uncertain mathematical characteristics of the dynamic responses of stochastic and interval Hamiltonian systems can be obtained by the symplectic algorithms, respectively. Then, the comparative study of calculation results by the symplectic homotopy perturbation method of stochastic and interval Hamiltonian systems is carried out and their compatible relationship is discussed. Eventually, three numerical examples are used to demonstrate the validity and engineering applicability in structural dynamic systems of the proposed method. The numerical results show the superiority of the proposed method in accuracy, stability and symplectic conservative compared with the non-symplectic Runge–Kutta method.