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Codimension-two bifurcations analysis and tracking control on a discrete epidemic model

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Abstract

In this paper, the dynamic behaviors of a discrete epidemic model with a nonlinear incidence rate obtained by Euler method are discussed, which can exhibit the periodic motions and chaotic behaviors under the suitable system parameter conditions. Codimension-two bifurcations of the discrete epidemic model, associated with 1:1 strong resonance, 1:2 strong resonance, 1:3 strong resonance and 1:4 strong resonance, are analyzed by using the bifurcation theorem and the normal form method of maps. Moreover, in order to eliminate the chaotic behavior of the discrete epidemic model, a tracking controller is designed such that the disease disappears gradually. Finally, numerical simulations are obtained by the phase portraits, the maximum Lyapunov exponents diagrams for two different varying parameters in 3-dimension space, the bifurcation diagrams, the computations of Lyapunov exponents and the dynamic response. They not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.

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Authors and Affiliations

  1. Institute of Systems Science, Northeastern University, Shenyang, 110004, China

    Na Yi & Qingling Zhang

  2. School of Sciences, Liaoning Shihua University, Fushun, 113001, China

    Na Yi

  3. School of Management, Shenyang University of Technology, Shenyang, 110870, China

    Peng Liu

  4. Department of Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Yanping LIN

Authors
  1. Na Yi
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  2. Qingling Zhang
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  3. Peng Liu
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  4. Yanping LIN
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Corresponding author

Correspondence to Qingling Zhang.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant Nos. 60974004 and 71001074, and the Science Research Foundation of Department of Education of Liaoning Province of China under Grant No. W2010302.

This paper was recommended for publication by Editor Jinhu LÜ.

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Yi, N., Zhang, Q., Liu, P. et al. Codimension-two bifurcations analysis and tracking control on a discrete epidemic model. J Syst Sci Complex 24, 1033–1056 (2011). https://doi.org/10.1007/s11424-011-9041-0

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  • DOI: https://doi.org/10.1007/s11424-011-9041-0

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