Asymptotic expansions of unstable (stable) manifolds in time-discrete systems

1999/10/18 by Shin‐itiro Goto, Shin-itiro Goto, Goto, Shin-itiro +2
Computer Science · Mathematics · Physics and Astronomy · #Chaos control and synchronization #Chaos-based Image/Signal Encryption #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Dynamics and Fractals #chao-dyn #nlin.CD

paper · pdf · doi:10.48550/arxiv.chao-dyn/9910025

14pages, 7figures

arxiv created 1999/10/18 · openalex publication_date 1999/10/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Hénon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Hénon map.

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