Circuits, Systems, and Signal Processing

, Volume 35, Issue 6, pp 1933–1948 | Cite as

Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form

  • Sifeu Takougang KingniEmail author
  • Viet-Thanh Pham
  • Sajad Jafari
  • Guy Richard Kol
  • Paul Woafo


A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.


Three-dimensional autonomous chaotic system Circular equilibrium Stability analysis Circuit implementation Fractional-order system Chaos synchronization 



S.T.K. and G.R.K. thank Prof. Elisabeth Ngo Bum, the Director of the Institute of Mines and Petroleum Industries (University of Maroua, Cameroon), for creating a good environment with perfect balance between teaching and research time.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sifeu Takougang Kingni
    • 1
    • 5
    • 6
    Email author
  • Viet-Thanh Pham
    • 2
  • Sajad Jafari
    • 3
  • Guy Richard Kol
    • 1
    • 4
  • Paul Woafo
    • 5
  1. 1.Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum IndustriesUniversity of MarouaMarouaCameroon
  2. 2.School of Electronics and TelecommunicationsHanoi University of Science and TechnologyHanoiVietnam
  3. 3.Department of Biomedical EngineeringAmirkabir University of TechnologyTehranIran
  4. 4.School of Geology and Mining EngineeringUniversity of NgaoundéréNgaoundéréCameroon
  5. 5.Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of ScienceUniversity of Yaoundé IYaoundéCameroon
  6. 6.Applied Physics Research Group (APHY)Vrije Universiteit BrusselBrusselsBelgium

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