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High-Order Computer-Assisted Estimates of Topological Entropy


A Dissertation

In partial fulfillment of the requirements for a degree of Doctor of Philosophy from Michigan State University.


Abstract

The concept of Taylor Models is introduced, which offers highly accurate C zero estimates for the enclosures of functional dependencies, combining high-order Taylor polynomial approximation of functions and rigorous estimates of the truncation error, performed using verified interval arithmetic. The focus of this work is on the application of Taylor Models in algorithms for strongly nonlinear dynamical systems.

A method to obtain sharp rigorous enclosures of Poincaré maps for certain types of flows and surfaces is developed and numerical examples are presented.

Differential algebraic techniques allow the efficient and accurate computation of polynomial approximations for invariant curves of certain planar maps around hyperbolic fixed points. Subsequently we introduce a procedure to extend these polynomial curves to verified Taylor Model enclosures of local invariant manifolds with C zero errors of size 10E-10 -- 10E-14, and proceed to generate the global invariant manifold tangle up to comparable accuracy through iteration in Taylor Model arithmetic.

Knowledge of the global manifold structure up to finite iterations of the local manifold pieces enables us to find all homoclinic and heteroclinic intersections in the generated manifold tangle. Combined with the mapping properties of the homoclinic points and their ordering we are able to construct a subshift of finite type as a topological factor of the original planar system to obtain rigorous lower bounds for its topological entropy. This construction is fully automatic and yields homoclinic tangles with several hundred homoclinic points.

As an example rigorous lower bounds for the topological entropy of the Héenon map are computed, which to the best knowledge of the authors yield the largest such estimates published so far.


J. Grote (2008)


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