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Chaos and Bifurcation in Controllable Jerk-Based Self-Excited Attractors

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Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 133))

Abstract

In the recent decades, utilization of chaotic systems has flourished in various engineering applications. Hence, there is an increasing demand on generalized, modified and novel chaotic systems. This chapter combines the general equation of jerk-based chaotic systems with simple scaled discrete chaotic maps. Two continuous chaotic systems based on jerk-equation and discrete maps with scaling parameters are presented. The first system employs the scaled tent map, while the other employs the scaled logistic map. The effects of different parameters on the type of the response of each system are investigated through numerical simulations of time series, phase portraits, bifurcations and Maximum Lyapunov Exponent (MLE) values against all system parameters. Numerical simulations show interesting behaviors and dependencies among these parameters. Analogy between the effects of the scaling parameters is presented for simple one-dimensional discrete chaotic systems and the continuous jerk-based chaotic systems with more complicated dynamics. The impacts of these scaling parameters appear on the effective ranges of other main system parameters and the ranges of the obtained solution. The dependence of equilibrium points on the sign of one of the scaling parameters results in coexisting attractors according to the signs of the parameter and the initial point. In addition, switching can be used to generate double-scroll attractors. Moreover, bifurcation and chaos are studied for fractional-order of the derivative.

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Authors and Affiliations

  1. Faculty of Engineering, Engineering Mathematics and Physics Department, Cairo University, Giza, 12613, Egypt

    Wafaa S. Sayed & Ahmed G. Radwan

  2. Nanoelectronics Integrated Systems Center, Nile University, Cairo, 12588, Egypt

    Ahmed G. Radwan

  3. Faculty of Engineering, Electronics and Communications Engineering Department, Cairo University, Giza, 12613, Egypt

    Hossam A. H. Fahmy

Authors
  1. Wafaa S. Sayed
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  2. Ahmed G. Radwan
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  3. Hossam A. H. Fahmy
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Corresponding author

Correspondence to Wafaa S. Sayed .

Editor information

Editors and Affiliations

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

    Viet-Thanh Pham

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

    Sundarapandian Vaidyanathan

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

    Christos Volos

  4. Division of Dynamics, Faculty of Mechanical Engineering, Lodz University of Technology, Łódź, Poland

    Tomasz Kapitaniak

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Sayed, W.S., Radwan, A.G., Fahmy, H.A.H. (2018). Chaos and Bifurcation in Controllable Jerk-Based Self-Excited Attractors. In: Pham, VT., Vaidyanathan, S., Volos, C., Kapitaniak, T. (eds) Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors. Studies in Systems, Decision and Control, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-71243-7_2

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  • DOI: https://doi.org/10.1007/978-3-319-71243-7_2

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