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Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map

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  • Yao, Xiao-Yue
  • Li, Xian-Feng
  • Jiang, Jun
  • Leung, Andrew Y.T.

Abstract

This paper devotes to a detailed bifurcation analysis of a two-dimensional non-invertible map, obtained using a symmetric coupling between one-dimensional logistic maps. The critical normal form coefficients method is employed to detect bifurcations and to explore further critical conditions without explicit reduction to the center manifold. The results show that the two-dimensional map undergoes codimension-one (codim-1) bifurcations such as transcritical, pitchfork, period-doubling, Neimark–Sacker, and codim-2 bifurcations including transcritical-flip, pitchfork-flip, strong resonances 1:2, 1:3, 1:4. For each bifurcation, the critical normal form coefficients are calculated to check the non-degeneracy conditions and predict the bifurcation scenarios around the bifurcation points. To validate the theoretical results, all bifurcation curves of fixed points are plotted with the aid of the numerical continuation method. Weak resonances are also specified by the isoclines on the bi-parameter plane. The results will help in understanding the occurrence and the structure of bifurcation cascades observed in many coupled discrete systems.

Suggested Citation

  • Yao, Xiao-Yue & Li, Xian-Feng & Jiang, Jun & Leung, Andrew Y.T., 2022. "Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s096007792200830x
    DOI: 10.1016/j.chaos.2022.112651
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    References listed on IDEAS

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    1. Zhang, Ying-Qian & He, Yi & Wang, Xing-Yuan, 2018. "Spatiotemporal chaos in mixed linear–nonlinear two-dimensional coupled logistic map lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 148-160.
    2. Gancio, Juan & Rubido, Nicolás, 2022. "Critical parameters of the synchronisation's stability for coupled maps in regular graphs," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    3. Elsadany, A.A. & Yousef, A.M. & Elsonbaty, Amr, 2018. "Further analytical bifurcation analysis and applications of coupled logistic maps," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 314-336.
    4. Alidousti, J. & Eskandari, Z. & Avazzadeh, Z., 2020. "Generic and symmetric bifurcations analysis of a three dimensional economic model," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    5. Parsamanesh, Mahmood & Erfanian, Majid, 2021. "Stability and bifurcations in a discrete-time SIVS model with saturated incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    6. Li, Xian-Feng & Leung, Andrew Y.T. & Jiang, Jun, 2018. "Synchronizability and mode-locking of two scaled quadratic maps via symmetric direct-coupling," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 239-247.
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