Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems | Zendy

Shinitiro Goto | Zendy; Kazuhiro Nozaki | Zendy

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Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

Author(s) -

Shinitiro Goto,

Kazuhiro Nozaki

Publication year - 2001

Publication title -

progress of theoretical physics

Language(s) - English

Resource type - Journals

eISSN - 1347-4081

pISSN - 0033-068X

DOI - 10.1143/ptp.105.99

Subject(s) - homoclinic orbit , attractor , physics , dissipative system , fixed point , manifold (fluid mechanics) , stable manifold , asymptotic expansion , renormalization , fractal , hyperbolic equilibrium point , periodic point , mathematical analysis , simple (philosophy) , hénon map , mathematical physics , chaotic , mathematics , bifurcation , nonlinear system , quantum mechanics , hyperbolic manifold , hyperbolic function , computer science , mechanical engineering , philosophy , epistemology , artificial intelligence , engineering

By means of an updated renormalization method, we construct asymptoticexpansions for unstable manifolds of hyperbolic fixed points in the double-wellmap and the dissipative H\'enon map, both of which exhibit the stronghomoclinic chaos. In terms of the asymptotic expansion, a simple formulation ispresented to give the first homoclinic point in the double-well map. Even atruncated expansion of the unstable manifold is shown to reproduce thewell-known many-leaved (fractal) structure of the strange attractor in theH\'enon map.

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