Bifurcation structure of an invariant three-torus and its computational sensitivity generated in a three-coupled delayed logistic map

• Hidaka Shuya

  Department of Electronics and Bioinfomatics, Meiji University

• Inaba Naohiko

  Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University

• Kamiyama Kyohei

  Department of Electronics and Bioinfomatics, Meiji University

• Sekikawa Munehisa

  Department of Mechanical and Intelligent Engineering, Utsunomiya University

• Endo Tetsuro

  Department of Electronics and Bioinfomatics, Meiji University

抄録

This study investigates an invariant three-torus (IT₃) and related quasi-periodic bifurcations generated in a three-coupled delayed logistic map. The IT₃ generated in this map corresponds to a four-dimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasi-periodic oscillations, we demonstrate that it is necessary that the number of iterations deleted as a transient state is large and similar to the number of iterations to be averaged as a stationary state. For example, it is necessary to delete 10,000,000 transient iterations to appropriately evaluate the Lyapunov exponents if they are averaged for 10,000,000 stationary state iterations in this higher-dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by analyzing a graph of the Lyapunov exponents, we show that the local bifurcation transition from an invariant two-torus(IT₂) to an IT₃ is caused by a Neimark-Sacker bifurcation, which can be a one-dimension-higher quasi-periodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT₂ to an IT₃ can be generated by a saddle-node bifurcation, which can be denoted as a one-dimension-higher quasi-periodic saddle-node bifurcation. Furthermore, we find a codimension-two bifurcation point at which the curves of a QH bifurcation and a quasi-periodic saddle-node bifurcation intersect.

収録刊行物

• Nonlinear Theory and Its Applications, IEICE

  Nonlinear Theory and Its Applications, IEICE 6 (3), 433-442, 2015

  一般社団法人 電子情報通信学会

キーワード

• invariant three-torus
• bifurcations
• delayed logistic map
• computational sensitivity

詳細情報

• CRID: 1390282680322196224
• NII論文ID: 130005085583
• DOI: 10.1587/nolta.6.433
• ISSN: 21854106
• Web Site: https://www.jstage.jst.go.jp/article/nolta/6/3/6_433/_pdf
• 本文言語コード: en
• データソース種別:
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• 抄録ライセンスフラグ: 使用不可