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Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems

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  • Fiaz, Muhammad
  • Aqeel, Muhammad
  • Marwan, Muhammad
  • Sabir, Muhammad

Abstract

In this article we studied a 3D autonomous system derived from Sprot B, C, Van der Schrier-Mass and Munmuangsaen Srisuchinwong chaotic systems for existence of zero Hopf bifurcation with the help of averaging theory of first order. Fractional order analysis of the derived system are discussed for stability of equilibrium points, chaotification condition, sensitivity dependence, Lyapunov exponents, Kaplan-Yorke dimension, chaotic time history and phase portraits. Novelty of the paper is investigation of integer and fractional order synchronization of derived system with famous Lorenz model by active control method under the same parametric values and initial conditions. By taking example of the model under consideration we generalized the synchronization for a class of integer and fractional order systems. We concluded that if a couple of integer order chaotic dynamical system is synchronized then its fractional order version will also be synchronized for same parametric values and initial conditions and vice versa. We also compared three different numerical techniques for synchronization. By calculating CPU timing for synchronization we determined that the integer order chaotic system was synchronized earlier than that of fractional order. The results so achieved show that it is sufficient to get synchronization of an integer order system if its fractional version also exists. This investigation contributes to minimize the cost of control for a class of dynamical systems when such control is made through synchronization. Numerical simulations are also provided to authenticate the analytical results.

Suggested Citation

  • Fiaz, Muhammad & Aqeel, Muhammad & Marwan, Muhammad & Sabir, Muhammad, 2022. "Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    • Handle: RePEc:eee:chsofr:v:155:y:2022:i:c:s0960077921010973
    DOI: 10.1016/j.chaos.2021.111743
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    References listed on IDEAS

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    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    2. Li, Chunguang & Chen, Guanrong, 2004. "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 55-61.
    3. Agrawal, S.K. & Srivastava, M. & Das, S., 2012. "Synchronization of fractional order chaotic systems using active control method," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 737-752.
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    1. Louiza Diabi & Adel Ouannas & Amel Hioual & Giuseppe Grassi & Shaher Momani, 2025. "The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization," Mathematics, MDPI, vol. 13(2), pages 1-24, January.

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