Existence of Invariant Cones in General 3-Dim Homogeneous Piecewise Linear Differential Systems with Two Zones

Abstract

Existence and number of invariant cones in general 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane are investigated. Implicit parametric expressions of two proper half slope maps whose intersections determine the existence and number of invariant cones are obtained. Based on these expressions, some sufficient conditions for the existence of at most three invariant cones are provided, and it is proved that the maximum number of invariant cones for some special cases is equal to 1 plus the maximum number of limit cycles in planar piecewise linear systems with a straight line separation. Moreover, it is illustrated by a numerical example with four invariant cones that the maximum number of invariant cones is not less than four. Specially, the main results provide a method to completely solve the existence and number of invariant cones in any specific 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane by using numerical method.