The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold

作者：Li, Shuangbao^*; Gong, Xiaojun; Zhang, Wei; Hao, Yuxin

来源：Nonlinear Dynamics, 2017, 89(2): 939-953.

DOI：10.1007/s11071-017-3493-2

• Melnikov method
• Planar piecewise-smooth systems
• Homoclinic orbits
• Switching manifolds
• Reset maps

In this paper, we extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth system subjected to a time-periodic perturbation. In this class, we suppose there exists a switching manifold with a more general form such that the plane is divided into two zones, and the dynamics in each zone is governed by a smooth system. Furthermore, we assume that the unperturbed system is a general planar piecewise-smooth system with non-zero trace and possesses a piecewise-smooth homoclinic orbit transversally crossing the switching manifold. We also define a reset map to describe the instantaneous impact rule on the switching manifold when a trajectory arrives at the switching manifold. Through a series of geometrical analysis and perturbation techniques, we obtain a Melnikov-type function to measure the separation of the unstable manifold and stable manifold under the effect of the time-periodic perturbations and the reset map. Finally, we use the presented Melnikov function to study global bifurcations and chaotic dynamics for a concrete planar piecewise-linear oscillator.

• 出版日期2017-7
• 单位北京工业大学; 中国民航大学