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A Three-Dimensional Chaotic System with Square Equilibrium and No-Equilibrium

  • Viet-Thanh PhamEmail author
  • Sundarapandian Vaidyanathan
  • Christos K. Volos
  • Sajad Jafari
  • Tomas Gotthans
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 688)

Abstract

Recently, Leonov and Kuznetsov have introduced a new definition “hidden attractor”. Systems with hidden attractors, especially chaotic systems, have attracted significant attention. Some examples of such systems are systems with a line equilibrium, systems without equilibrium or systems with stable equilibria etc. In some interesting new research, systems in which equilibrium points are located on different special curves are reported. This chapter introduces a three-dimensional autonomous system with a square-shaped equilibrium and without equilibrium points. Therefore, such system belongs to a class of systems with hidden attractors. The fundamental dynamics properties of such system are studied through phase portraits, Poincaré map, bifurcation diagram, and Lyapunov exponents. Anti-synchronization scheme for our systems is proposed and confirmed by the Lyapunov stability. Moreover, an electronic circuit is implemented to show the feasibility of the mathematical model. Finally, we introduce the fractional order form of such system.

Keywords

Chaos Hidden attractor No-equilibrium Square equilibrium Lyapunov exponents Bifurcation Synchronization Circuit SPICE 

Notes

Acknowledgements

Research described in this paper was supported by Czech Ministry of Education in frame of National Sustainability Program under grant GA15-22712S. V.-T. Pham is grateful to Le Thi Van Thu, Philips Electronics—Vietnam, for her help.

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Authors and Affiliations

  • Viet-Thanh Pham
    • 1
    Email author
  • Sundarapandian Vaidyanathan
    • 2
  • Christos K. Volos
    • 3
  • Sajad Jafari
    • 4
  • Tomas Gotthans
    • 5
  1. 1.School of Electronics and TelecommunicationsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Research and Development CentreVel Tech UniversityTamil NaduIndia
  3. 3.Physics DepartmentAristotle University of ThessalonikiThessalonikiGreece
  4. 4.Biomedical Engineering DepartmentAmirkabir University of TechnologyTehranIran
  5. 5.Department of Radio ElectronicsBrno University of TechnologyBrnoCzech Republic

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