Global analysis of limit cycles in networks of oscillators

Abstract

This paper is concerned with the global stability of limit cycle oscillations for a particular class of systems and networks. In previous work, we defined a class of parameter-dependent nonlinear systems exhibiting an almost globally asymptotically stable limit cycle. The results were proven for values of the parameter in the vicinity of a bifurcation value. In the present paper we restrict ourselves to a piecewise linear version of this class of systems and adapt numerical tools recently proposed in the literature to prove global stability of the limit cycle for a fixed value of the parameter above the bifurcation value. Furthermore, we show how the global stability results for one isolated oscillator are useful to prove the existence of a globally synchrone oscillation in particular networks of identical oscillators.