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Complex dynamics in a linear impulsive system

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  • Jiang, Guirong
  • Yang, Qigui

Abstract

The dynamical behavior of a linear impulsive system is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the equilibrium and period-one solution, the discontinuous jumps of eigenvalues, and the conditions for system possessing infinite period-two, period-three, and period-six solutions. By using discrete maps, the conditions of existence for Neimark–Sacker bifurcation are derived. In particular, chaotic behavior in the sense of Marotto’s definition of chaos is rigorously proven. Moreover, some detailed numerical results of the phase portraits, the periodic solutions, the bifurcation diagram, and the chaotic attractors, which are illustrated by some interesting examples, are in good agreement with the theoretical analysis.

Suggested Citation

  • Jiang, Guirong & Yang, Qigui, 2009. "Complex dynamics in a linear impulsive system," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2341-2353.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:5:p:2341-2353
    DOI: 10.1016/j.chaos.2008.09.012
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    References listed on IDEAS

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    1. Wang, Fengyan & Hao, Chunping & Chen, Lansun, 2007. "Bifurcation and chaos in a Monod type food chain chemostat with pulsed input and washout," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 826-839.
    2. Marotto, F.R., 2005. "On redefining a snap-back repeller," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 25-28.
    3. Jiang, Guirong & Lu, Qishao & Qian, Linning, 2007. "Complex dynamics of a Holling type II prey–predator system with state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 448-461.
    4. Gakkhar, Sunita & Negi, Kuldeep, 2008. "Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 626-638.
    5. Wang, Fengyan & Hao, Chunping & Chen, Lansun, 2007. "Bifurcation and chaos in a Tessiet type food chain chemostat with pulsed input and washout," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1547-1561.
    6. Wang, Fengyan & Hao, Chunping & Chen, Lansun, 2007. "Bifurcation and chaos in a Monod–Haldene type food chain chemostat with pulsed input and washout," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 181-194.
    7. Wu, Xiaoqun & Lu, Jun-an & Tse, Chi K. & Wang, Jinjun & Liu, Jie, 2007. "Impulsive control and synchronization of the Lorenz systems family," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 631-638.
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    1. Huo, Hai-Feng & Li, Wan-Tong, 2009. "Dynamics of continuous-time bidirectional associative memory neural networks with impulses and their discrete counterparts," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2218-2229.

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