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A new Lorenz-type hyperchaotic system with a curve of equilibria

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  • Chen, Yuming
  • Yang, Qigui

Abstract

Little seems to be known about hyperchaotic systems with a curve of equilibria. Based on the classical Lorenz system, this paper proposes a new four-dimensional Lorenz-type hyperchaotic system which has a curve of equilibria. This new system can generate not only hyperchaotic attractors but also chaotic, quasi-periodic and periodic attractors, as well as singular degenerate heteroclinic cycles. Of particular interest is the observation that there are four types of coexisting attractors of this new hyperchaotic system: (i) chaotic attractor and quasi-periodic attractor, (ii) chaotic attractor and singular degenerate heteroclinic cycle, (iii) periodic attractor and singular degenerate heteroclinic cycle, and (iv) different periodic attractors. Furthermore, many singular degenerate heteroclinic cycles are found, which may lead to complex dynamics of hyperchaotic system with a curve of equilibria.

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  • Chen, Yuming & Yang, Qigui, 2015. "A new Lorenz-type hyperchaotic system with a curve of equilibria," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 112(C), pages 40-55.
    • Handle: RePEc:eee:matcom:v:112:y:2015:i:c:p:40-55
    DOI: 10.1016/j.matcom.2014.11.006
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    References listed on IDEAS

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    1. Jafari, Sajad & Sprott, J.C., 2013. "Simple chaotic flows with a line equilibrium," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 79-84.
    2. Mahmoud, Gamal M. & Mahmoud, Emad E., 2010. "Synchronization and control of hyperchaotic complex Lorenz system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(12), pages 2286-2296.
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    Cited by:

    1. Laarem, Guessas, 2021. "A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it’s fractional order model and chaos sy," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Wang, Haijun & Dong, Guili, 2019. "New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 272-286.
    3. Kingni, Sifeu Takougang & Pham, Viet-Thanh & Jafari, Sajad & Woafo, Paul, 2017. "A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 209-218.
    4. Xiong Wang & Viet-Thanh Pham & Christos Volos, 2017. "Dynamics, Circuit Design, and Synchronization of a New Chaotic System with Closed Curve Equilibrium," Complexity, Hindawi, vol. 2017, pages 1-9, February.
    5. Pham, Viet–Thanh & Jafari, Sajad & Volos, Christos & Fortuna, Luigi, 2019. "Simulation and experimental implementation of a line–equilibrium system without linear term," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 213-221.
    6. Singh, Jay Prakash & Roy, Binoy Krishna, 2018. "Five new 4-D autonomous conservative chaotic systems with various type of non-hyperbolic and lines of equilibria," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 81-91.
    7. Yu Liu & Yan Zhou & Biyao Guo, 2023. "Hopf Bifurcation, Periodic Solutions, and Control of a New 4D Hyperchaotic System," Mathematics, MDPI, vol. 11(12), pages 1-14, June.
    8. Singh, Jay Prakash & Roy, B.K., 2016. "The nature of Lyapunov exponents is (+, +, −, −). Is it a hyperchaotic system?," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 73-85.
    9. Wang, Haijun & Li, Xianyi, 2018. "A novel hyperchaotic system with infinitely many heteroclinic orbits coined," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 5-15.
    10. Singh, Jay Prakash & Roy, Binoy Krishna & Jafari, Sajad, 2018. "New family of 4-D hyperchaotic and chaotic systems with quadric surfaces of equilibria," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 243-257.
    11. Pham, Viet–Thanh & Jafari, Sajad & Volos, Christos & Kapitaniak, Tomasz, 2016. "A gallery of chaotic systems with an infinite number of equilibrium points," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 58-63.

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