Global properties of continuous piecewise linear vector fields. Part II: Simplest symmetric case in ℝ2

Abstract

AbstractAmong non‐linear vector fields, the simplest that can be studied are those which are continuous and piecewise linear. Among these non‐linear vector fields a large and important subset are those vector fields which are odd symmetric. Associated with these types of vector fields are partitions of the state space into a finite number of regions. In each region the vector field is linear. On the boundary between regions it is required that the vector field be continuous from both regions in which it is linear. This presentation is devoted to the analysis in two dimensions of the simplest possible types of continuous piecewise linear vector fields with odd symmetry, namely those vector fields possessing a pair of symmetric boundary conditions.As a practical concern the analysis will attempt to ask and answer questions raised about the existence of steady state solutions. Since the local theory of fixed points in a linear vector field is sufficient to determine the stability of fixed points in a piecewise linear vector field, most of the steady state behaviour to be studied will be towards limit cycles. the results will present sufficient conditions for the existence, or non‐existence as the case may be, of limit cycles. Particular attention will be paid to the domain of attraction whenever possible.With these results qualitative statements may be made for piecewise linear models of physical systems which have odd symmetry.