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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Effect of Fractional-Order on the Dynamic of two Mutually Coupled van der Pol Oscillators: Hubs, Multistability and its Control

Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 83--98 | DOI:10.5890/DNC.2020.03.007

Ngo Mouelas Adèle$^{1}$, Kammogne Soup Tewa Alain$^{1}$, Kengne Romanic$^{1}$, Fotsin Hilaire Bertrand$^{1}$, Essimbi Zobo Bernard$^{2}$

$^{1}$ Laboratory of Condensed Matter, Electronics and Signal Processing (LAMACETS), University of Dschang, P.O. Box 67, Dschang, Cameroon

$^{2}$ Electronics Laboratory, University of Yaoundé 1, P.O. Box 812, Yaoundé, Cameroon

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Abstract

This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is fixed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponent are exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multistability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.

Acknowledgments

The authors thank the reviewers for their expertise and Dr Lekeufack Martin for his assistance in reading this manuscript.

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