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The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees

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  • Rao, Xiao-Bo
  • Zhao, Xu-Ping
  • Chu, Yan-Dong
  • Zhang, Jian-Gang
  • Gao, Jian-She

Abstract

We report the topology of stable periodic solutions of an SIR epidemic dynamics model with impulsive vaccination control, a hybrid discrete/continuous non-smooth system, in the parameter plane spanned by the pulse period T and the quantity of pulse vaccination b. This mode-locking topology is governed by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations rather than the fast-slow time scales, and its periodicity is in perfect agreement with the so-called Stern-Brocot sum tree, a derived tree from “Farey algorithm”. More surprisingly, the mode-locking order is unfolded in a way that never encountered before to emerge organized according to the reciprocal of the Stern-Brocot sum tree. Furthermore, the global organization of mode-locking is not isolated structure but an infinite Stern-Brocot sum tree cascade when tuning the control parameter T. The results obtained contribute a new framework to classify mode-locking oscillations observed in the hybrid system.

Suggested Citation

  • Rao, Xiao-Bo & Zhao, Xu-Ping & Chu, Yan-Dong & Zhang, Jian-Gang & Gao, Jian-She, 2020. "The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    • Handle: RePEc:eee:chsofr:v:139:y:2020:i:c:s096007792030429x
    DOI: 10.1016/j.chaos.2020.110031
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    References listed on IDEAS

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    1. Zeng, Guang Zhao & Chen, Lan Sun & Sun, Li Hua, 2005. "Complexity of an SIR epidemic dynamics model with impulsive vaccination control," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 495-505.
    2. Gallas, Jason A.C., 1994. "Dissecting shrimps: results for some one-dimensional physical models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 202(1), pages 196-223.
    3. Wiggers, Vinícius & Rech, Paulo C., 2017. "Multistability and organization of periodicity in a Van der Pol–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 103(C), pages 632-637.
    4. Rao, Xiao-Bo & Chu, Yan-Dong & Chang, Ying-Xiang & Zhang, Jian-Gang, 2018. "Broken Farey tree and fractal in a hexagonal centrifugal governor with a spring," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 251-255.
    5. Shaw, Pankaj Kumar & Chaubey, Neeraj & Mukherjee, S. & Janaki, M.S. & Iyengar, A.N. Sekar, 2019. "A continuous transition from chaotic bursting to chaotic spiking in a glow discharge plasma and its associated long range correlation to anti correlation behaviour," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 126-134.
    6. Anderson Hoff & Juliana Santos & Cesar Manchein & Holokx Albuquerque, 2014. "Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
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