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Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order

Author

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  • Coronel-Escamilla, A.
  • Gómez-Aguilar, J.F.
  • Torres, L.
  • Escobar-Jiménez, R.F.
  • Valtierra-Rodríguez, M.

Abstract

In this paper, we propose a state-observer-based approach to synchronize variable-order fractional (VOF) chaotic systems. In particular, this work is focused on complete synchronization with a so-called unidirectional master–slave topology. The master is described by a dynamical system in state-space representation whereas the slave is described by a state observer. The slave is composed of a master copy and a correction term which in turn is constituted of an estimation error and an appropriate gain that assures the synchronization. The differential equations of the VOF chaotic system are described by the Liouville–Caputo and Atangana–Baleanu–Caputo derivatives. Numerical simulations involving the synchronization of Rössler oscillators, Chua’s systems and multi-scrolls are studied. The simulations show that different chaotic behaviors can be obtained if different smooths functions defined in the interval (0,1] are used as the variable order of the fractional derivatives. Furthermore, simulations show that the VOF chaotic systems can be synchronized.

Suggested Citation

  • Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & Torres, L. & Escobar-Jiménez, R.F. & Valtierra-Rodríguez, M., 2017. "Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 487(C), pages 1-21.
    • Handle: RePEc:eee:phsmap:v:487:y:2017:i:c:p:1-21
    DOI: 10.1016/j.physa.2017.06.008
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    Cited by:

    1. Dubey, Ved Prakash & Kumar, Rajnesh & Kumar, Devendra, 2019. "Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 762-772.
    2. Eshaghi, Shiva & Khoshsiar Ghaziani, Reza & Ansari, Alireza, 2020. "Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 321-340.
    3. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
    4. Peng, Yuexi & Sun, Kehui & Peng, Dong & Ai, Wei, 2019. "Dynamics of a higher dimensional fractional-order chaotic map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 96-107.
    5. Vázquez-Guerrero, P. & Gómez-Aguilar, J.F. & Santamaria, F. & Escobar-Jiménez, R.F., 2020. "Design of a high-gain observer for the synchronization of chimera states in neurons coupled with fractional dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    6. Owolabi, Kolade M., 2019. "Mathematical modelling and analysis of love dynamics: A fractional approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 849-865.
    7. Pratap, A. & Raja, R. & Cao, J. & Rihan, Fathalla A. & Seadawy, Aly R., 2020. "Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Chen, Yun & Xu, Yanyi & Lin, Qian & Zhang, Xiyong, 2020. "Model and criteria on the global finite-time synchronization of the chaotic gyrostat systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 515-533.
    9. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.

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