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High Period Fixed Points, Stable and Unstable Manifolds, and Chaos in Accelerator Transfer Maps

Abstract

In this paper we present an algorithm for a verified global fixed point finder. More specifically, a method is described to automatically identify and classify regions of the search space which are guaranteed to either contain none, precisely one, or one or more fixed points, as well as regions that may or may not contain fixed points. The fixed point finder is implemented with Taylor Models in COSY INFINITY, allowing for very efficient identification of fixed points even in numerically complicated systems with high dependency and strong cancellation.

We then apply the fixed point finder to find higher order periodic points in a transfer map taken from the Tevatron accelerator. The results are compared to predictions made from tune shifts computed using normal form theory. A high order approximation to the stable and unstable manifolds of a set of hyperbolic periodic points is computed and shown.

A. Wittig, M. Berz, Vestnik Mathematica 10,2 (2014) 93-110

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