# Global Properties of Continuous Piecewise-Linear Vector Fields Part II: Simplest Symmetric Case in R^2

### R. Lum and Leon O. Chua

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M90/29

1990

Among nonlinear vector fields, the simplest of which can be studied are those which are continuous and piecewise linear. Among these nonlinear 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 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 nonexistence as the case may be, for 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.

BibTeX citation:

@techreport{Lum:M90/29, Author = {Lum, R. and Chua, Leon O.}, Title = {Global Properties of Continuous Piecewise-Linear Vector Fields Part II: Simplest Symmetric Case in R^2}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1458.html}, Number = {UCB/ERL M90/29}, Abstract = {Among nonlinear vector fields, the simplest of which can be studied are those which are continuous and piecewise linear. Among these nonlinear 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 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 nonexistence as the case may be, for 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.} }

EndNote citation:

%0 Report %A Lum, R. %A Chua, Leon O. %T Global Properties of Continuous Piecewise-Linear Vector Fields Part II: Simplest Symmetric Case in R^2 %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/ERL M90/29 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1458.html %F Lum:M90/29