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This paper is devoted to the study of an averaging principle for fractional stochastic differential equations in Rn with Lévy motion, using an integral transform method. We obtain a time-averaged effective equation under suitable assumptions. Furthermore, we show that the solutions of the averaged equation approach the solutions of the original equation. Our results provide a better understanding for effective approximation of fractional dynamical systems with non-Gaussian Lévy noise.

This paper presents an averaging principle for fractional stochastic differential equations in ℝn with fractional order 0 < α < 1. We obtain a time-averaged equation under suitable conditions, such that the solutions to original fractional equation can be approximated by solutions to simpler averaged equation. By mathematical manipulations, we show that the mild solution of two equations before and after averaging are equivalent in the sense of mean square, which means the classical Khasminskii approach for the integer order systems can be extended to fractional systems.

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It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.