Sufficient conditions and criteria for the pulse-shaped explosion related to equilibria in a class of nonlinear systems

Abstract

The pulse-shaped explosion (PSE) is a novel sharp transition behavior which can give rise to bursting oscillations. Exploring a general theory applied to PSE is an important problem in the research of PSE. In this paper, we propose a two-dimensional ordinary differential equation (ODE), which can represent a class of dynamical systems with multiple-frequency slow excitations, and contribute to the theoretical understanding of the generation of PSE. Based on the fast–slow analysis, we firstly transform the ODE with multiple-frequency slow excitations into the one with a single slow variable, and thus we can obtain the fast subsystem, based on which the sufficient conditions for the existence of PSE related to equilibria are given. Then, we analyze the sufficient conditions for PSE by considering the behavior of the equilibrium theoretically. Besides, the criteria of different PSE patterns related to equilibria are developed to investigate different manifestations of PSE. Finally, the validity of the theoretical analysis is showed by numerical results of several concrete examples satisfying the class of ODE given in this paper. Our results provide the sufficient conditions and criteria for PSE involved in the proposed ODE family, and facilitate one to generate any desired dynamical systems exhibiting PSE within the framework of the suggested equation form.