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Chaos in the Newton–Leipnik system with fractional order

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Listed:
  • Sheu, Long-Jye
  • Chen, Hsien-Keng
  • Chen, Juhn-Horng
  • Tam, Lap-Mou
  • Chen, Wen-Chin
  • Lin, Kuang-Tai
  • Kang, Yuan

Abstract

The dynamics of fractional-order systems has attracted increasing attention in recent years. In this paper, the dynamics of the Newton–Leipnik system with fractional order was studied numerically. The system displays many interesting dynamic behaviors, such as fixed points, periodic motions, chaotic motions, and transient chaos. It was found that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order for this system to yield chaos is 2.82. A period-doubling route to chaos in the fractional-order system was also found.

Suggested Citation

  • Sheu, Long-Jye & Chen, Hsien-Keng & Chen, Juhn-Horng & Tam, Lap-Mou & Chen, Wen-Chin & Lin, Kuang-Tai & Kang, Yuan, 2008. "Chaos in the Newton–Leipnik system with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 98-103.
    • Handle: RePEc:eee:chsofr:v:36:y:2008:i:1:p:98-103
    DOI: 10.1016/j.chaos.2006.06.013
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    References listed on IDEAS

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    1. Li, Chunguang & Chen, Guanrong, 2004. "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 55-61.
    2. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    3. Wang, Xuedi & Tian, Lixin, 2006. "Bifurcation analysis and linear control of the Newton–Leipnik system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 31-38.
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    Cited by:

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    3. Yifan Zhang & Tianzeng Li & Zhiming Zhang & Yu Wang, 2022. "Novel Methods for the Global Synchronization of the Complex Dynamical Networks with Fractional-Order Chaotic Nodes," Mathematics, MDPI, vol. 10(11), pages 1-22, June.
    4. Agrawal, S.K. & Srivastava, M. & Das, S., 2012. "Synchronization of fractional order chaotic systems using active control method," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 737-752.
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    6. Baishya, Chandrali & Premakumari, R.N. & Samei, Mohammad Esmael & Naik, Manisha Krishna, 2023. "Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
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    9. Luo, Chuanwen & Wang, Gang & Wang, Chuncheng & Wei, Junjie, 2009. "A new interpretation of chaos," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1294-1300.

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