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Impulsive stabilization of chaos in fractional-order systems

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Abstract

This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos suppression by stabilizing the unstable system equilibria is provided. A nonexistence result of periodic solutions is presented, and the commensurate fractional-order Lorenz system is simulated for illustration.

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Notes

  1. The fractional order has been chosen close to 1, such that chaos is as strong as possible (note that the minimum order for the fractional-order Lorenz system to be chaotic is \(q=0.99\) [37]).

  2. Once the trajectory enters the sphere, it remains inside.

  3. Some related works will be published elsewhere later.

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Acknowledgements

The authors thank to Professor Julien Clinton Sprott for interesting discussions related to the energy approach. M.-F. Danca is supported by Tehnic B SRL. M. Fečkan is supported in part by the Slovak Research and Development Agency under the Contract No. APVV-14-0378 and by the Slovak Grant Agency VEGA Nos. 2/0153/16 and 1/0078/17. G. Chen is supported by the Hong Kong Research Grants Council under the GRF Grant CityU 11234916.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Avram Iancu University of Cluj-Napoca, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute of Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

  3. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava, Slovak Republic

    Michal Fečkan

  4. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovak Republic

    Michal Fečkan

  5. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China

    Guanrong Chen

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  1. Marius-F. Danca
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Correspondence to Marius-F. Danca.

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Danca, MF., Fečkan, M. & Chen, G. Impulsive stabilization of chaos in fractional-order systems. Nonlinear Dyn 89, 1889–1903 (2017). https://doi.org/10.1007/s11071-017-3559-1

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