Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations

The purpose of this paper is to report a novel route to mixed-mode oscillations (MMOs), i.e., the mechanism which we call speed escape of attractors, based on a Rayleigh equation with multiple-frequency excitations. The fast subsystem exhibits two critical points, which bound the region of a periodic attractor or an equilibrium point attractor and outside which are divergent regions. We show that both the periodic attractor and equilibrium point attractor can rapidly move far away from the original place when the control parameter reaches the critical values. This helps us reveal the novel mechanism to MMOs, and two different types of MMOs, i.e., MMOs of point-point type and MMOs of cycle-cycle type, are thus obtained. Besides, the effects of excitations on the MMOs are explored. We show that the amplitudes and frequencies of excitations may have important influences on MMOs. Our results enrich the possible routes to MMOs as well as the underlying mechanisms of MMOs.