Classification on Boundary-Equilibria and Singular Continuums of Continuous Piecewise Linear Systems

In this paper, we show that any switching hypersurface of n-dimensional continuous piecewise linear systems is an (n - 1)-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.

Automated summary

In this paper, it is shown that any switching hypersurface of n-dimensional continuous piecewise linear systems is an (n - 1)-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, local phase portraits and indices near the boundary equilibria and singular continuum between two parallel switching lines are presented. The existence of multiple boundary-equilibria and singular continuums with many parallel switching lines is also demonstrated.