Rapid Deterioration of Convergence in Taylor Expansions of Linearizing Maps of Hénon Maps at Hyperbolic Fixed Points

Koichi Hiraide and Chihiro Matsuoka ()
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Koichi Hiraide: Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan University, Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
Chihiro Matsuoka: Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan University, Sugimoto, Sumiyoshi, Osaka 558-8585, Japan

Mathematics, 2025, vol. 13, issue 21, 1-17

Abstract: In this paper, we prove that the Taylor expansions of analytic functions appearing in the linearization of quadratic maps at hyperbolic fixed points do not successfully approximate invariant manifolds, such as stable and unstable manifolds, when higher-order terms are truncated. This fact was pointed out by Newhouse et al. in their numerical experiments, and implies that the Taylor expansions are inadequate for quantitatively studying dynamical systems such as quadratic maps. In fact, it is shown that the computational complexity for the approximations by the Taylor expansions grows exponentially.

Keywords: discrete dynamical systems; Hénon maps; linearization; Taylor expansions; hyperbolic fixed points; computational complexity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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