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Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique

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Abstract

This research deals with a comparative numerical analysis of chaos in two systems with non-integer derivatives. The one-scroll system and circle equilibrium system with different hidden attractors are simulated considering the fractal derivative, Khalil and Atangana conformable derivatives, and the truncated M-derivative considering a constant and variable-order. Phase portraits are shown, as well as bifurcation diagrams, and Lyapunov exponents are obtained. Later, 0–1 test, dynamic death analysis, and sensitivity to initial conditions are considered to choose which derivative produces richer chaotic behaviors. According to those mentioned above, we could observe that the M-derivative not only generalizes Khalil’s type conformable derivative but also its two non-integer orders produce interesting dynamic behaviors compared to the remaining derivatives. In the numerical results, we observe that the variable order makes the system more sensitive to the change in the initial conditions. The new chaotic behaviors with constant and variable order are used to develop a synchronization scheme of two identical one-scroll systems via the backstepping technique with the truncated M-derivative involved.

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia for founding this work through Research Groups program under Grant Number (R.G.P1./278/42).

Funding

King Khalid University Researcher Supporting Project Number (R.G.P1./278/42).

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Authors and Affiliations

  1. Escuela Nacional de Estudios Superiores Unidad Juriquilla, Universidad Nacional, Autónoma de México, Boulevard Juriquilla 3001, Juriquilla La Mesa, 76230, Juriquilla, QRO, México

    J. E. Solís-Pérez

  2. Université de Technologie de Compiègne, CNRS, UMR 7253 Heudiasyc, Compiègne Cedex, France

    J. Betancourt-Vera

  3. CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, MOR, México

    J. F. Gómez-Aguilar

  4. Department of Mathematics, Faculty of Science, King Khalid University, Abha, 61413, Kingdom of Saudi Arabia

    A. A. Alderremy

  5. Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

    Shaban Aly

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  1. J. E. Solís-Pérez
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  2. J. Betancourt-Vera
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  3. J. F. Gómez-Aguilar
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Contributions

JES: conceptualization, methodology, validation, investigation, Writing-original draft preparation, writing-review; JB-V: conceptualization, methodology, validation, writing—original draft preparation, and writing—review; JFG-A: conceptualization, methodology, validation, and writing–review and editing; AAA: validation, formal analysis, and investigation; SA: validation, formal analysis, and investigation. All authors have read and agreed to the published version of the manuscript.

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Correspondence to J. F. Gómez-Aguilar.

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Solís-Pérez, J.E., Betancourt-Vera, J., Gómez-Aguilar, J.F. et al. Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique. Eur. Phys. J. Spec. Top. 231, 1931–1968 (2022). https://doi.org/10.1140/epjs/s11734-022-00468-y

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