Complex Bifurcation of Arnol’d Tongues Generated in Three-Coupled Delayed Logistic Maps

  • Daiki OgusuEmail author
  • Shuya Hidaka
  • Naohiko Inaba
  • Munehisa Sekikawa
  • Tetsuro Endo
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 191)


This study investigates quasi-periodic bifurcations and Arnol’d resonance webs generated in a three-coupled delayed logistic map. Complex bifurcation structure is generated when a conventional Arnol’d tongue transits to a higher-dimensional Arnol’d tongue. We discovered that, at least, two periodic attractors coexist in the conventional Arnol’d tongue which can bifurcate to two one-tori via doubly-folded Neimark–Sacker bifurcation.


Periodic Solution Bifurcation Curve Bifurcation Structure Periodic Attractor Sacker Bifurcation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Broer, H., Simó, C., Vitolo, R.: The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol’d resonance web. Bull. Belg. Math. Soc. Simon Stevin 15, 769–787 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Vitolo, R., Broer, H., Simó, C.: Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regul. Chaot. Dyn. 16, 154–184 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Takens, F., Wagener, F.O.O.: Resonances in skew and reducible quasi-periodic Hopf bifurcations. Nonlinearity 13, 377–396 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kuznetsov, Y.A., Meijer, H.G.E.: Remarks on interacting Neimark–Sacker bifurcations. J. Differ. Equ. Appl. 12, 1009–1035 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sekikawa, M., Inaba, N., Kamiyama, K., Aihara, K.: Three-dimensional tori and Arnold tongues. Chaos 24, 013017 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Broer, H., Simó, C., Vitolo, R.: Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance ‘bubble’. Physica D 237, 1773–1799 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61, 1605–1616 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Daiki Ogusu
    • 1
    Email author
  • Shuya Hidaka
    • 1
  • Naohiko Inaba
    • 2
  • Munehisa Sekikawa
    • 3
  • Tetsuro Endo
    • 1
  1. 1.Department of Electronics and BioinformaticsMeiji UniversityKawasakiJapan
  2. 2.Organization for the Strategic Coordination of Research and Intellectual PropertyMeiji UniversityKawasakiJapan
  3. 3.Department of Mechanical and Intelligent EngineeringUtsunomiya UniversityUtsunomiyaJapan

Personalised recommendations