Self-induced synchronization by large delay

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