On the Number of Invariant Cones and Existence of Periodic Orbits in 3-dim Discontinuous Piecewise Linear Systems

Abstract

For a family of discontinuous 3-dim homogeneous piecewise linear dynamical systems with two zones, we investigate the number of invariant cones and the existence of periodic orbits as a spatial relationship between the invariant manifolds of the subsystem changes. By studying the number of real roots of a quadratic equation induced by slopes of half straight lines starting from the origin in required domain, we obtain complete results on the number and stability of invariant cones. Especially, we prove that the maximum number of invariant cones is two, and obtain complete parameter regions on which there exist one or two invariant cones, on which one or two fake cones (corresponding to real roots of the quadratic equation that are not in the required domain) appear and on which an invariant cone will be foliated by periodic orbits.