The most probable response of some prototypical stochastic nonlinear dynamical systems

Abstract

Abstract The response of the stochastic system hardly provides useful information for researching its characteristic. Therefore, the paper is aimed at utilizing a deterministic tool, namely, the most probable response, to explore the stochastic nonlinear system. Firstly, one defines the most probable response of the stochastic system. Then, its analytical solution is derived by incorporating the extremum theory into the associated Fokker–Planck equation. Finally, two examples are given to illustrate respectively the implication of this method. Meanwhile, it can conclude that the large the intensity of multiplicative noise is, the more obvious the impact is on the response of nonlinear system. However, no matter how the intensity of additive noise changes, the most probable response does not change. The numerical method verifies the validity of analytical solution.