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 non-isolated equilibria) between two parallel switching lines. The definition of the index of singular continuum is introduced. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.
2010 Mathematics Subject Classification. Primary 34C05; 34A26; 49J52.