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A Three-Dimensional No-Equilibrium Chaotic System: Analysis, Synchronization and Its Fractional Order Form

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Fractional Order Control and Synchronization of Chaotic Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 688))

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Abstract

Recently, a new classification of nonlinear dynamics has been introduced by Leonov and Kuznetsov, in which two kinds of attractors are concentrated, i.e. self-excited and hidden ones. Self-excited attractor has a basin of attraction excited from unstable equilibria. So, from that point of view, most known systems, like Lorenz’s system, Rössler’s system, Chen’s system, or Sprott’s system, belong to chaotic systems with self-excited attractors. In contrast, a few unusual systems such as those with a line equilibrium, with stable equilibria, or without equilibrium, are classified into chaotic systems with hidden attractor. Studying chaotic system with hidden attractors has become an attractive research direction because hidden attractors play an important role in theoretical problems and engineering applications. This chapter presents a three-dimensional autonomous system without any equilibrium point which can generate hidden chaotic attractor. The fundamental dynamics properties of such no-equilibrium system are discovered by using phase portraits, Lyapunov exponents, bifurcation diagram, and Kaplan–Yorke dimension. Chaos synchronization of proposed systems is achieved and confirmed by numerical simulation. In addition, an electronic circuit is implemented to evaluate the theoretical model. Finally, fractional-order form of the system with no equilibrium is also investigated.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.02–2012.27

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Authors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam

    Viet-Thanh Pham, Thang Manh Hoang & Vu Van Yem

  2. Research and Development Centre, Vel Tech University, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos K. Volos

  4. Faculty of Computers and Information, Benha University, Benha, Egypt

    Ahmad Taher Azar

  5. Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt

    Ahmad Taher Azar

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  1. Viet-Thanh Pham
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  6. Vu Van Yem
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Correspondence to Viet-Thanh Pham .

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  1. Faculty of Computers and Information, Benha University Faculty of Computers and Information, AND Nanoelectronics Integrated Systems Center (NISC), Nile University, Egypt. , Benha, Egypt

    Ahmad Taher Azar

  2. Research and Development Centre, Vel Tech University Research and Development Centre, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Department of Mathematics and CS, University of Tebessa Department of Mathematics and CS, Tebessa, Algeria

    Adel Ouannas

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Pham, VT., Vaidyanathan, S., Volos, C.K., Azar, A.T., Hoang, T.M., Van Yem, V. (2017). A Three-Dimensional No-Equilibrium Chaotic System: Analysis, Synchronization and Its Fractional Order Form. In: Azar, A., Vaidyanathan, S., Ouannas, A. (eds) Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol 688. Springer, Cham. https://doi.org/10.1007/978-3-319-50249-6_15

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