05-246 A. Haro, R. de la Llave

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical implementation and examples. (14210K, Gzipped postscript) Jul 15, 05

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Abstract. In this paper we describe the implementation of the numerical algorithms for the computation of invariant manifolds (both tori and their whiskers) in quasi-periodically forced systems presented in the companion paper mp_arc 04-350. The algorithms are based on the parameterization method introduced in mp_arc 04-348 for this type of systems. We apply the implemented algorithms to some examples considered already in the literature and report on efficiency, accuracy, storage requirements, running times, etc. The new methods allow us to continue invariant objects close to the breakdown of their hyperbolicity properties. We find that some of the systems loose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. Computing several measures of hyperbolicity (the distance between the invariant bundles, and the Lyapunov multipliers) we find power laws with universal exponents. We also observe that, even if the rigorous justifications in mp_arc 04-348 are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with non-orientable bundles, which we compute.

Files: 05-246.src( 05-246.keywords , examples7.ps.gz.mm )