Rigorous High-Precision Enclosures of Fixed Points and Their Invariant Manifolds

A Dissertation

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

Abstract

The well established concept of Taylor Models is introduced, which offer highly accurate $C^{0}$ enclosures of functional dependencies, combining high-order polynomial approximation of functions and rigorous estimates of the truncation error, performed using verified arithmetic. The focus of this work is on the application of Taylor Models in algorithms for strongly non-linear dynamical systems.

A method is proposed to extend the existing implementation of Taylor Models in COSY INFINITY from double precision coefficients to arbitrary precision coefficients. Great care is taken to maintain the highest efficiency possible by adaptively adjusting the precision of higher order coefficients in the polynomial expansion. High precision operations are based on clever combinations of elementary floating point operations yielding exact values for round-off errors. An experimental high precision interval data type is developed and implemented. Algorithms for the verified computation of intrinsic functions based on the High Precision Interval datatype are developed and described in detail. The application of these operations in the implementation of High Precision Taylor Models is discussed.

An application of Taylor Model methods to the verification of fixed points is presented by verifying the existence of a period 15 fixed point in a near standard Hénon map. Verification is performed using different verified methods such as double precision Taylor Models, High Precision intervals and High Precision Taylor Models. Results and performance of each method are compared.

An automated rigorous fixed point finder is implemented, allowing the fully automated search for all fixed points of a function within a given domain. It returns a list of verified enclosures of each fixed point, optionally verifying uniqueness within these enclosures. An application of the fixed point finder to the rigorous analysis of beam transfer maps in accelerator physics is presented.

Previous work done by Johannes Grote is extended to compute very accurate polynomial approximations to invariant manifolds of discrete maps of arbitrary dimension around hyperbolic fixed points. The algorithm presented allows for automatic removal of resonances occurring during construction.

A method for the rigorous enclosure of invariant manifolds of continuous systems is introduced. Using methods developed for discrete maps, polynomial approximations of invariant manifolds of hyperbolic fixed points of ODEs are obtained. These approximations are outfit with a sharp error bound which is verified to rigorously contain the manifolds. While we focus on the three dimensional case, verification in higher dimensions is possible using similar techniques.

Integrating the resulting enclosures using the verified COSY VI integrator, the initial manifold enclosures are expanded to yield sharp enclosures of large parts of the stable and unstable manifolds. To demonstrate the effectiveness of this method, we construct enclosures of the invariant manifolds of the Lorenz system and show pictures of the resulting manifold enclosures. To the best of our knowledge, these enclosures are the largest verified enclosures of manifolds in the Lorenz system in existence.

This dissertation is also available as a reprint (PDF) that has been reformatted for readability. Minor corrections have been made to the content of the reprint, as listed in the preface of the reprint.

A. Wittig (2011)

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