Lyapunov-type numbers and torus breakdown: numerical aspects and a case study

Abstract

In this work we examine the breakdown mechanism for a 2-torus associated to a system of coupled oscillators. To this end, we rely on computation of the Lyapunov-type numbers, as defined by Fenichel in N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971) 193–226. We give some general results on these numbers, and some specific constructive results for the case in which there is phase-locking on the torus. This is the situation for the system of coupled oscillators we consider, and it leads to great simplifications in the computation of the Lyapunov-type numbers.