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Chaos in a generalized van der Pol system and in its fractional order system

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  • Ge, Zheng-Ming
  • Hsu, Mao-Yuan

Abstract

In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system.

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  • Ge, Zheng-Ming & Hsu, Mao-Yuan, 2007. "Chaos in a generalized van der Pol system and in its fractional order system," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1711-1745.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:5:p:1711-1745
    DOI: 10.1016/j.chaos.2006.03.028
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    References listed on IDEAS

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    1. dos Santos, Angela M. & Lopes, Sergio R. & Viana, R.L.Ricardo L., 2004. "Rhythm synchronization and chaotic modulation of coupled Van der Pol oscillators in a model for the heartbeat," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 338(3), pages 335-355.
    2. Ge, Z.-M. & Cheng, J.-W., 2005. "Chaos synchronization and parameter identification of three time scales brushless DC motor system," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 597-616.
    3. Ge, Zheng-Ming & Lee, Ching-I, 2005. "Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1855-1864.
    4. Ge, Zheng-Ming & Chen, Yen-Sheng, 2005. "Adaptive synchronization of unidirectional and mutual coupled chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 881-888.
    5. Ahmad, Wajdi M., 2005. "Hyperchaos in fractional order nonlinear systems," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1459-1465.
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    Cited by:

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    2. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    3. Liang Chen & Chengdai Huang & Haidong Liu & Yonghui Xia, 2019. "Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order," Mathematics, MDPI, vol. 7(6), pages 1-16, June.
    4. Zhang, Weiwei & Zhou, Shangbo & Li, Hua & Zhu, Hao, 2009. "Chaos in a fractional-order Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1684-1691.
    5. Yu, Yongguang & Li, Han-Xiong, 2008. "The synchronization of fractional-order Rössler hyperchaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1393-1403.

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