On Network Perturbations of Electrical Circuits and Singular Perturbation of Dynamical Systems

Abstract

Dynamical systems of electrical networks including jumping behavior have been reduced locally to singular perturbation theory for ordinary differential equations. This chapter aims to give the foundation of a general theory of dynamical systems arising from electrical circuits which involves the theory of relaxation oscillations. It examines the phenomenon by means of N. Fenichel’s geometric singular perturbation theory of dynamical systems. F. Takens reduces the study to the consideration of singular perturbations of constrained differential equations, in the case when a suitable real-valued function exists. As an application, the jumping behaviors of states of electrical networks are given in neighborhoods of compact subsets of regular domains. S. S. Sastry and C. A. Desoer treat jump behavior related to singular perturbations of constrained differential equations as models of the jump behavior of electrical circuits. The chapter reduces the situations globally to the geometric singular perturbation theory of N. Fenichel.