Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system

Abstract

In this paper, a double pendulum model with multi-point collision is established to study the sub-harmonic bifurcation of high-dimensional coupled non-smooth systems. Considering the coupling and non-smoothness of the system, a two-step decoupling method is proposed to detect the sub-harmonic bifurcation of a two-degree-of-freedom non-smooth coupled system. The core view is to introduce energy-time scale transformation to overcome the obstacle of the system coupled term. In the first step, a reversible transformation is introduced to decouple the system. This transformation enables the coupled form of the impact term, which presents novel obstacles to the high-dimensional non-smooth system. By introducing energy-time scale transformation in the second step, the system is expressed as a smooth decoupling form of the energy coordinate, and the trouble of impact term coupled is solved. Furthermore, the sub-harmonic Melnikov function which depends on frequency, amplitude of excitation and impact recovery coefficient is derived by using the two-step decoupling method. Hence, the sub-harmonic Melnikov function is extended to the high-dimensional non-smooth system, which reveals the influence of the impact recovery coefficient on the existence of sub-harmonic periodic orbits. The innovation of this method is that it solves the coupled problem of non-smooth terms, quantifies the impact of impact recovery coefficient on the dynamic behavior of the system, and provides a theoretical basis for the actual parameter design and control in engineering. The obtained theoretical results are verified through the numerical simulations.