Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems
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. I...
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| author | A. E. Matouk T. N. Abdelhameed D. K. Almutairi M. A. Abdelkawy M. A. E. Herzallah |
| author_facet | A. E. Matouk T. N. Abdelhameed D. K. Almutairi M. A. Abdelkawy M. A. E. Herzallah |
| author_sort | A. E. Matouk |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T09:35:09Z |
| format | Article |
| id | doaj.art-1bcc44333d774fac811929c2ebd857c8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:35:09Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1bcc44333d774fac811929c2ebd857c82023-11-16T17:21:46ZengMDPI AGMathematics2227-73902023-01-0111359110.3390/math11030591Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing SystemsA. E. Matouk0T. N. Abdelhameed1D. K. Almutairi2M. A. Abdelkawy3M. A. E. Herzallah4Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi ArabiaDepartment of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi ArabiaCollege of Engineering, Majmaah University, Al-Majmaah 11952, Saudi ArabiaMathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, EgyptFaculty of Science, Zagazig University, Zagazig 44519, EgyptThis 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.https://www.mdpi.com/2227-7390/11/3/591integer-order MAVPD systemfractional-orderchaosself-excited attractorshidden attractors |
| spellingShingle | A. E. Matouk T. N. Abdelhameed D. K. Almutairi M. A. Abdelkawy M. A. E. Herzallah Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems Mathematics integer-order MAVPD system fractional-order chaos self-excited attractors hidden attractors |
| title | Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems |
| title_full | Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems |
| title_fullStr | Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems |
| title_full_unstemmed | Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems |
| title_short | Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems |
| title_sort | existence of self excited and hidden attractors in the modified autonomous van der pol duffing systems |
| topic | integer-order MAVPD system fractional-order chaos self-excited attractors hidden attractors |
| url | https://www.mdpi.com/2227-7390/11/3/591 |
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