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The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

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  • Wu, Wenjuan
  • Chen, Zengqiang
  • Yuan, Zhuzhi

Abstract

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.

Suggested Citation

  • Wu, Wenjuan & Chen, Zengqiang & Yuan, Zhuzhi, 2009. "The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2340-2356.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:5:p:2340-2356
    DOI: 10.1016/j.chaos.2007.07.016
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    References listed on IDEAS

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    Cited by:

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    3. Du Mingjing & Yulan Wang, 2019. "Some Novel Complex Dynamic Behaviors of a Class of Four-Dimensional Chaotic or Hyperchaotic Systems Based on a Meshless Collocation Method," Complexity, Hindawi, vol. 2019, pages 1-15, October.
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    5. Wu, Wen-Juan & Chen, Zeng-Qiang & Yuan, Zhu-Zhi, 2009. "A computer-assisted proof for the existence of horseshoe in a novel chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2756-2761.

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