Presentation 2013-05-28
Pattern Categorization of the Bifurcation of Quasi-Periodic Solutions by using Covariant Lyapunov Bundle
Kyohei KAMIYAMA, Motomasa KOMURO, Tetsuro ENDO,
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Abstract(in English) In continuous dynamical system, a periodic orbit becomes a fix point by taking a certain Poincare section. Therefore, by calculating the eigenvalues and eigenvectors of the fixed point, we can calculate local stability and progressing direction. In contrast, a quasi-periodic solution (two-torus) becomes an invariant closed curve (ICC) in Poincare section. For an ICC one can calculate Lyapunov exponents which are generalization of the eigenvalues of a fixed point. However, there is no defined value corresponding to the eigenvectors of a fixed point. In this paper, we define covariant Lyapunov vector (CLV) at each mapped point of ICC, and succeed to draw a bundle of CLV along the ICC. This is a part of tangent bundle formed by tangent space of ICC. Therefore, we call it "covariant Lyapunov bundle." In this research, we classify various local bifurcations of ICC by using covariant Lyapunov vectors and Lyapunov exponents. As examples, we introduce the behavior of CLB for saddle-node bifurcation, pitchfork bifurcation, period-doubling bifurcation of type 1 (increase of winding number), that of type 2 (collapse of stability), and Neimark-Sacker bifurcation by using continuous dynamical systems.
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Keyword(in English) covariant Lyapunov bundle / covariant Lyapunov vector / Lyapunov exponent / invariant closed curve / quasi-periodic solution / local bifurcation
Paper # NLP2013-23
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Committee NLP
Conference Date 2013/5/20(1days)
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Registration To Nonlinear Problems (NLP)
Language JPN
Title (in Japanese) (See Japanese page)
Sub Title (in Japanese) (See Japanese page)
Title (in English) Pattern Categorization of the Bifurcation of Quasi-Periodic Solutions by using Covariant Lyapunov Bundle
Sub Title (in English)
Keyword(1) covariant Lyapunov bundle
Keyword(2) covariant Lyapunov vector
Keyword(3) Lyapunov exponent
Keyword(4) invariant closed curve
Keyword(5) quasi-periodic solution
Keyword(6) local bifurcation
1st Author's Name Kyohei KAMIYAMA
1st Author's Affiliation Electronics and Bioinfomatics, Meiji University()
2nd Author's Name Motomasa KOMURO
2nd Author's Affiliation Center for Fundamental Education, Teikyo University of Science
3rd Author's Name Tetsuro ENDO
3rd Author's Affiliation Electronics and Bioinfomatics, Meiji University
Date 2013-05-28
Paper # NLP2013-23
Volume (vol) vol.113
Number (no) 69
Page pp.pp.-
#Pages 4
Date of Issue