KAM tori and whiskered invariant tori for non-autonomous systems

Abstract

We consider non-autonomous dynamical systems which converge to autonomous (or periodic) systems exponentially fast in time. Such systems appear naturally as models of many physical processes affected by external pulses.We introduce definitions of non-autonomous invariant tori and non-autonomous whiskered tori and their invariant manifolds and we prove their persistence under small perturbations, smooth dependence on parameters and several geometric properties (if the systems are Hamiltonian, the tori are Lagrangian manifolds). We note that such definitions are problematic for general time-dependent systems, but we show that they are unambiguous for systems converging exponentially fast to autonomous.The proof of persistence relies only on a standard Implicit Function Theorem in Banach spaces and it does not require that the rotations in the tori are Diophantine nor that the systems we consider preserve any geometric structure. We only require that the autonomous system preserves these objects. In particular, when the autonomous system is integrable, we obtain the persistence of tori with rational rotational. We also discuss fast and efficient algorithms for their computation. The method also applies to infinite dimensional systems which define a good evolution, e.g. PDE’s.When the systems considered are Hamiltonian, we show that the time dependent invariant tori are isotropic. Hence, the invariant tori of maximal dimension are Lagrangian manifolds. We also obtain that the (un)stable manifolds of whiskered tori are Lagrangian manifolds.We also include a comparison with the more global theory developed in Blazevski and de la Llave (2011).