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Chaos Synchronization of the Modified Autonomous Van der Pol-Duffing Circuits via Active Control

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Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 3

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this work, we study the dynamics and synchronization of a chaotic system describes the Modified Autonomous Van der Pol-Duffing (MAVPD) circuit. The detailed bifurcation diagrams are given to show the rich dynamics of the proposed system. Lyapunov exponents are calculated to verify the existence of chaos in this system. Chaos synchronization of MAVPD system is obtained using active control method. According to the qualitative theory of fractional differential equations, the existence and uniqueness of solutions for a class of commensurate fractional-order MAVPD systems are investigated. Furthermore, based on the stability theory of fractional-order systems, the conditions of local stability of linear fractional-order system are discussed. Moreover, the existence of chaotic behaviors in the fractional-order MAVPD system is shown. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three. Phase synchronization of the fractional-order MAVPD system is also achieved using an active control technique. Numerical simulations show the effectiveness of the proposed synchronization schemes.

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Acknowledgements

The authors wish to thank Prof. E. Ahmed and Prof. H.N. Agiza for discussion and help. The corresponding author ‘Matouk’ wishes to thank the editor, Dr. S. Banerjee, and Dr. Eid AL-Haisouni, the Dean of Prep-Year college for their kind help and support.

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Authors and Affiliations

  1. Faculty of Science, Mathematics Department, Mansoura University, Mansoura, 35516, Egypt

    Ahmed Sadek Hegazi & Ahmed Ezzat Matouk

  2. College of Prep Year, Mathematics Department, Hail University, Hail, 2440, Saudi Arabia

    Ahmed Ezzat Matouk

Authors
  1. Ahmed Sadek Hegazi
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  2. Ahmed Ezzat Matouk
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Correspondence to Ahmed Ezzat Matouk .

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Editors and Affiliations

  1. , Institute for Mathematical Research, University Putra Malaysia, Hak Cipta Terpelihara UPM 2007, Serdang, 43400, Malaysia

    Santo Banerjee

  2. , Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, Torino, 10129, Italy

    Lamberto Rondoni

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Hegazi, A.S., Matouk, A.E. (2013). Chaos Synchronization of the Modified Autonomous Van der Pol-Duffing Circuits via Active Control. In: Banerjee, S., Rondoni, L. (eds) Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 3. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34017-8_7

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  • DOI: https://doi.org/10.1007/978-3-642-34017-8_7

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  • Publisher Name: Springer, Berlin, Heidelberg

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  • Online ISBN: 978-3-642-34017-8

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