Stochastic averaging in parametric regions near separatrices of integrability

Abstract Stochastic averaging in Hamiltonian framework, as a powerful tool for the dimensional reduction of strongly-nonlinear stochastic dynamical systems, achieves its enormous efficacy by distinguishing the integrability and resonance of associated conservative systems. Its implementation and mathematical expressions of drift and diffusion coefficients are completely different for different categories. For multi-degree-of-freedom systems, a slight changing of stiffness coefficients may induce an immediate conversion of the integrability, and correspondingly, notably different mathematical expressions, which means a sudden changing of dynamical behaviors. This work is devoted to this anti-intuitive phenomenon through a two-degree-of-freedom nonlinear stochastic system with adjustable parameters, by introducing the concept of the degree of integrability, applying stochastic averaging for quasi-integrable/quasi-nonintegrable systems to the parametric region near the separatrix of integrability, and comparing their accuracy of prediction. Numerical results illustrate that there exists a specific band around the separatrix of integrability, as system parameters fall in which, stochastic averaging for quasi-integrable systems achieves higher accuracy than that for quasi-nonintegrable systems even though the system itself is non-integrable. This work uncovers the existence of such specific band, and constitutes a necessary supplement to stochastic averaging in Hamiltonian framework.