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

In this paper, we show that any switching hypersurface of [Formula: see text]-dimensional continuous piecewise linear systems is an [Formula: see text]-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.