The global dynamics of linear refracting systems of focus–node or center–node type

Abstract

We investigate the global dynamics of discontinuous planar piecewise linear refracting systems with a straight line of separation for focus–node (FN) type, center–node (CN) type, focus–improper node (FN′) type and center–improper node (CN′) type. By complete studying singularities, the invariant manifolds, limit cycles and all the possible configuration of orbits, we give the sufficient and necessary conditions of these system existing limit cycle for all parameters regions and prove that this four types of systems have at most one limit cycle. In addition, we also obtain that these refracting systems have 26 kinds of global topological phase portraits in the Poincaré disk.