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Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

Author

Listed:
  • A. E. Matouk

    (Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia
    College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia)

  • T. N. Abdelhameed

    (Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia
    College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia
    Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt)

  • D. K. Almutairi

    (College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia)

  • M. A. Abdelkawy

    (Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt
    Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia)

  • M. A. E. Herzallah

    (Faculty of Science, Zagazig University, Zagazig 44519, Egypt
    Faculty of Science and Humanities, Shaqra University, Al-Dawadmi 11911, Saudi Arabia)

Abstract

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.

Suggested Citation

  • A. E. Matouk & T. N. Abdelhameed & D. K. Almutairi & M. A. Abdelkawy & M. A. E. Herzallah, 2023. "Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems," Mathematics, MDPI, vol. 11(3), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:591-:d:1044134
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    References listed on IDEAS

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    1. Manuel Henriques & Duarte Valério & Paulo Gordo & Rui Melicio, 2021. "Fractional-Order Colour Image Processing," Mathematics, MDPI, vol. 9(5), pages 1-15, February.
    2. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    3. Kumar, Sunil & Kumar, Ranbir & Cattani, Carlo & Samet, Bessem, 2020. "Chaotic behaviour of fractional predator-prey dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    4. A. Othman Almatroud & A. E. Matouk & Wael W. Mohammed & Naveed Iqbal & Saleh Alshammari & Rigoberto Medina, 2022. "Self-Excited and Hidden Chaotic Attractors in Matouk’s Hyperchaotic Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2022, pages 1-14, March.
    5. Ajay Kumar & Sara Salem Alzaid & Badr Saad T. Alkahtani & Sunil Kumar, 2022. "Complex Dynamic Behaviour of Food Web Model with Generalized Fractional Operator," Mathematics, MDPI, vol. 10(10), pages 1-23, May.
    6. Denis de Carvalho Braga & Luis Fernando Mello & Marcelo Messias, 2009. "Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor," Mathematical Problems in Engineering, Hindawi, vol. 2009, pages 1-26, November.
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