Self-induced synchronization by large delay

Abstract

We investigate the dynamics of a delay differential equation obtained by perturbing a vector field f:Rn→Rn, admitting a stable periodic orbit, by using a delayed feedback control term ηg(x,x(t−τ)) of same regularity, where η is small and τ large so that ητ be bounded but non small. We prove that trajectories starting in a neighborhood (of size independent of the parameters η,τ) of this original periodic orbit enter asymptotically a periodic regime, and that the number of such distinct periodic regimes increases (almost) linearly when ητ increases infinitely. Our result is based on the construction of an invariant manifold via a process inspired by the Lyapunov-Perron method for integral operators associated to ordinary differential equations, and on the persistence of normally hyperbolic invariant manifolds for semi-flows on Banach spaces. The statement we provide here complements already known results on periodic orbits of delay differential equations.