Whiskered Tori for Presymplectic Dynamical Systems

Abstract

We prove persistence result of whiskered tori for the dynamical system which preserves an exact presymplectic form. The results are given in an a-posteriori format. Given an approximate solution of an invariance equation which satisfies some non-degeneracy assumptions, we conclude that there is a true solution close by. The proof is based on certain iterative procedure by which the accuracy of the approximate solutions of the invariance equation can be improved. The iterative procedure is not based on transformation theory, which is cumbersome for presymplectic systems, but on finding corrections to the solutions of the invariance equation. This iterative procedure takes advantage of identities that come from the preservation of the geometric structure and leads to a very efficient numerical method which has low storage requirements, low operator count per step and it is quadratically convergent. We note that a particular case of presymplectic systems is symplectic perturbed by quasi-periodic systems.