Conewise linear systems: a characterization of transitive cones in 3D-space

Abstract

A spatial (3D) piecewise linear system with multiple modes having conewise state spaces is considered. A single mode in this system is called transitive from one (respectively, two) of its borders if every trajectory that starts in its interior or at a border travels in its interior, hits that border (respectively, one of the two borders), and goes out of the cone. This paper characterizes transitive cones in case the dynamics in the cone is dictated by real and distinct eigenvalues. An example of a 3D piecewise linear system composed of transitive cones illustrates how a nonlinear oscillator can be synthesized.