Compound relaxation oscillations connected by pulse-shaped explosion

Abstract

Relaxation oscillations are ubiquitous in various fields of natural science and engineering technology. Exploring possible routes to relaxation oscillations is one of the important issues in the study of relaxation oscillations. Recently, the pulse-shaped explosion (PSE), a novel mechanism which can lead to relaxation oscillations, has been reported. The PSE means pulse-shaped sharp quantitative changes related the variation of system parameters in branches of equilibrium points and limit cycles, which leads the system’s trajectory to undertake sharp transitions and further induces relaxation oscillations. Regarding externally and parametrically excited nonlinear systems with different frequency ratios, some work on PSE has been reported. The present paper focuses on the PSE and the related relaxation oscillations in a externally and parametrically excited Mathieu-van der Pol-Duffing system. We show that if there is an initial phase difference between the slow excitations with the same frequency ratio, the fast subsystem may compose of two parts with different expressions, each of which determines a different vector field. As a result, the bistable behaviors are observed in the system. In particular, one of the vector fields exhibits two groups of bifurcation behaviors, which are symmetric with respect to the positive and negative PSE, and each can induce a cluster in the relaxation oscillations. Based on this, we present several routes to compound relaxation oscillations, and obtain new types of relaxation oscillations connected by pulse-shaped explosion, which we call compound “subHopf/fold-cycle” relaxation oscillations and compound “supHopf/supHopf” relaxation oscillations, respectively. Our results show that the two clusters in the resultant relaxation oscillations are connected by the PSE, and the initial phase difference plays an important role in transitions to different attractors and the generation of relaxation oscillations. Although the research in this paper is based on a specific nonlinear system, we would like to point out that the bistable behaviors, the PSE and the resultant compound relaxation oscillations can also be observed in other dynamical systems. The reason is that the fast subsystem composes of two different vector fields induced by the initial phase difference, which dose not rely on a specific system. The results of this paper deepen the research about PSE as well as the complex dynamics of relaxation oscillations.

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

“…We observe that the system shows the possibility of the development of periodic and chaotic bursting that appeared due to a sharp explosion in the equilibrium points of the system, known as a pulse-shaped explosion. Although the emergence of periodic compound bursting patterns via pulse-shaped explosion is reported, earlier [37,44,45], the generic nature of the pulse-shaped explosion induced bursting pattern is demonstrated in the chosen model. Also, interestingly, we provide evidence for the development of rare but recurrent chaotic large bursting along with regular small oscillations, which approved all the conditions of the EE phenomenon.…”

“…Two different types of regular bursting patterns, that is, bursting types of point-point type and cycle-cycle type, have been observed in the Rayleigh system via pulseshaped explosion [37]. In addition to that, the pulse-shaped explosion can also induce different bursting patterns in other dynamical systems as well [44,45]. Even though considerable work has been done in recent times regarding the pulse-shaped explosion-induced bursting dynamics, this area still requires further investigation.…”