Periodic solutions in systems of piecewise- linear differential equations

Abstract

Motivated by the periodic behaviour of regulatory networks within cell biology and neurology, we have studied the periodic solutions of piecewise-linear, first- order differential equations with identical relative decay rates. The flow of the solution trajectories can be represented qualitatively by a directed graph. By examining the cycles in this graph and solving the eigenvalue problem for corresponding mapping matrices, all closed, period-1 orbits can be found by analytical means. Theorems about their exist- ence, stabiiiiy and uniqueness are derived. For three-dimensional systems, the basins of attraction of the limit cycles can be explicitly determined and it is shown that higher periodic and chaotic solutions do not exist.