Compound relaxation oscillations influenced by non-smooth bifurcations in a Filippov Langford system

This paper aims to explore the dynamical mechanism of compound relaxation oscillations generated in the Filippov system, focusing on the influence of bifurcations induced by discontinuities on compound relaxation oscillations. A non-smooth Filippov system with two scales is built by introducing a non-smooth term and a external excitation into the Langford system based on the unfolding cusp-Hopf normal form. By specifying the slow variable as the bifurcation parameter, regular equilibrium bifurcation, boundary equilibrium bifurcation, and a variety of sliding bifurcations in the fast subsystem are obtained. Three types of oscillation modes under different parameter conditions are given through the numerical simulation. The dynamical mechanisms of each compound relaxation oscillation are also explained by combining the bifurcation analysis and slow–fast analysis. It is observed that persistence bifurcation realizes the stable transition between the pseudo-equilibrium branch and equilibrium branch. The grazing–sliding and switching–sliding bifurcations only change the topology of the limit cycle. None of these non-smooth bifurcations results in the transition between the quiescent state and spiking state, whereas the jumping behavior between different equilibrium branches may lead to the restructuring of the relaxation oscillation mode. In addition, a distinctive silent mode is observed, manifested by the complete disappearance of the compound relaxation oscillation.