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Zero-Hopf singularity in bidirectional ring network model with delay

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Abstract

This paper reports a bidirectional ring network model with delay. Zero-Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the zero-Hopf singularity and show that the model can exhibit pitchfork and Hopf bifurcation. Some numerical simulations are given to support the analytic results, and near the zero-Hopf singularity point, this model displays quasi-periodic, double periodic and multiple periodic trajectory.

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Acknowledgments

This work is supported by Fundamental Research Funds for the Central Universities (Project No. XDJK2014C118, SWU114007) and Natural Science Foundation of China (Grant no: 61374078), and also supported by the Natural Science Foundation Project of Chongqing CSTC (Grant no. cstc2014jcyjA40014). This publication was made possible by NPRP Grant No. NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

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Authors and Affiliations

  1. School of Electronics and Information Engineering, Southwest University, Chongqing, 400715, People’s Republic of China

    Xing He & Chuandong Li

  2. Texas A&M University at Qatar, Doha, 23874, Qatar

    Tingwen Huang

  3. School of Computer Science, Chongqing University of Education, Chongqing, 400067, China

    Junjian Huang

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  1. Xing He
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  2. Chuandong Li
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  3. Tingwen Huang
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  4. Junjian Huang
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Correspondence to Xing He.

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He, X., Li, C., Huang, T. et al. Zero-Hopf singularity in bidirectional ring network model with delay. Nonlinear Dyn 78, 2605–2616 (2014). https://doi.org/10.1007/s11071-014-1612-x

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  • DOI: https://doi.org/10.1007/s11071-014-1612-x

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