Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations

This paper investigates the mechanisms of small parametrical perturbations in controlling chaos in a class of non-autonomous piecewise smooth oscillators, which describe a large class of nonlinear dynamical systems in the real world. The analytical expressions of two homoclinic orbits of unperturbed piecewise smooth oscillators, which connect the same hyperbolic saddle point are solved analytically. Firstly, when there are no small parametrical perturbations, by using Melnikov׳s approach, it is rigorously proven that the homoclinic chaos in the Smale horseshoes sense exists when the system׳s parameters are selected above the threshold for chaos occurrence. Secondly, under the small parametrical perturbations, by using Melnikov׳s approach, a sufficient criterion is derived, serving as designing the parameters of the control signal, i.e., amplitude and phase position. In the process of computing Melnikov׳s functions, it is found that the expressions of Melnikov׳s functions could not be solved analytically because the homoclinic orbits are highly complicated. To this end, a numerical algorithm is proposed. Numerical simulations are presented to verify the theoretical results. The results of this paper can be used to explore the underlying chaotic behaviors of the inertial neural network model.