Systematic Computer-Assisted Proof of Branches of Stable Elliptic Periodic Orbits and Surrounding Invariant Tori

We present a concurrent algorithm for rigorous validation of the existence of continuous branches of stable elliptic fixed points for area-preserving planar maps. The method utilizes a classical theorem of Siegel and Moser combined with computed-assisted estimation of higher order derivatives of maps, continuation along the parameter range, and concurrent scheduling of tasks. We apply the algorithm to certain exemplary Poincaré maps coming from reversible or Hamiltonian systems: the periodically forced pendulum equations, the Michelson system, and the Hénon--Heiles Hamiltonian. Moreover, our algorithm provides at once a computer-assisted proof of the existence of wide branches of stable elliptic periodic solutions and the existence of invariant tori surrounding them.