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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


Dumitru Baleanu (editor)

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


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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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.


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


  1. [1]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New York.
  2. [2]  Borah, M. and Roy, B.K. (2018), Can fractional-order coexisting attractors undergo a rotational phenomenon?, ISA Transactions, 82, 2-17.
  3. [3]  Kammogne, S.T., Kengne, R., Ahmad, T.A., Sundarapandian, V., Fotsin, H.B. and Ngo, M.A. (2018), Dynamics analysis and synchronization in relay coupled fractional order colpitts oscillators, Advances in System Dynamics and Control, 11, 317-356.
  4. [4]  Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012), Fractional calculus models and numerical methods, Series on Complexity, Nonlinearity and Chaos, 3, 41-87
  5. [5]  Debnath, L. (2003), Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci, 54, 3413-3442.
  6. [6]  Borah, M., Singh, P.P., and Roy, B.K. (2016), Improved chaotic dynamics of a fractional order system, its chaossuppressed synchronization and circuit implementation, Circ. Syst. Signal. Process, 35, 1871-907.
  7. [7]  Ahmad,W.M. and Sprott, J.C. (2003), Chaos in fractional-order autonomous nonlinear systems, Chaos Solitons Fractals, 16, 339-351.
  8. [8]  Sun, K., Wang, X., and Sprott, J.C. (2010), Bifurcations and chaos in fractional order simplified Lorenz system, Int. J. Bifurc. Chaos, 20, 1209-1219.
  9. [9]  Li, C. and Peng, G. (2004), Chaos in Chen’s system with a fractional-order, Chaos Solitons Fractals, 22, 443-450.
  10. [10]  Lu, J.G.(2006), Chaotic dynamics of the fractional-order Lu system and its synchronization, Phys. Lett. A., 354, 305- 311.
  11. [11]  Li, C. and, Chen, G. (2004), Chaos and hyperchaos in the fractional-order Rössler equations, Phys. A, Stat. Mech. Appl. 341, 55-61.
  12. [12]  Lu, J.G. (2005), Chaotic dynamics and synchronization of fractional-orderArneodo’s systems, Chaos Solitons Fractals, 26, 1125-1133.
  13. [13]  Hartley, T.T., Lorenzo, C.F., and Qammer, H.K. (1995), Chaos in a fractional-order Chua’s system, IEEE Trans. Circuits. Syst., I, 42, 485-490.
  14. [14]  Arena, P., Caponetto, R., Fortuna, L., and Porto, D. (1997), Chaos in a fractional-orderDuffing system, In: Proceedings ECCTD, Budapest, 1259-1262.
  15. [15]  Sheu, L.J., Chen., H.K., Chen, J.H., Tam, L.M., Chen, W.C., and Lin, K.T. and Yuan, K. (2008), Chaos in the Newton- Leipnik system with fractional-order, Chaos Solitons Fractals, 36, 98-103.
  16. [16]  Kammogne, S.T.A., Ahmad, T.A., Kengne, R., and Fotsin, H.B. (2019), Stability analysis and robust sychronization of fractional-order modified colpitts oscillators, International Journal of Automation and Control (IJAAC), In press.
  17. [17]  Suchorsky, M.K. and Rand, R.H. (2012), A pair of van der Pol oscillators coupled by fractional derivatives, Nonlinear Dynamics, 69, 313-324.
  18. [18]  Kengne, R., Tchitnga, R., Kammogne, S.T., Grzegorz, L., Fomethe, A., and Li, C. (2018) Fractional-order twocomponent oscillator: stability and network synchronization using a reduced number of control signals, Eur. Phys. J. B., 91, 304-323, (Springer).
  19. [19]  Rompala, K., Rand, R., and Howland, H. (2007), Dynamics of three coupled van der Pol oscillators with application to circadian rhythms, Commun. Nonlinear Sci. Numer. Simulat, 12, 794-803.
  20. [20]  Kapitaniak, T. (1998), Chaos for engineers: Theory, applications and control (Springer, New York).
  21. [21]  Goska, A. and Krawiecki, A. (2006), Analysis of phase synchronization of coupled chaotic oscillators with empirical mode decomposition, Physical Review E, 74, 046217.
  22. [22]  Wen, S.F., Shen, Y.J., Yang S.P., and Wang, J. (2017), Dynamical response of Mathieu-Duffing oscillator with fractional-order delayed feedback, Chaos, Solitons & Fractals, 94, 54-62.
  23. [23]  Feng, X. and Xueyuan, L. (2009), Asymptotic solution of the van der Pol oscillator with small fractional damping, Physica Scripta, 2009, T136.
  24. [24]  Duraisamy, P. and Nandha, T.G. (2018), Existence and uniqueness of solutions for a coupled system of higher order fractional differential equations with integral boundary conditions, Discontinuity, Nonlinearity, and Complexity, 7, 1-14.
  25. [25]  Li, S., Niu, J., and Li, X. (2018), Primary resonance of fractional-order Duffing-van der Pol oscillator by harmonic balance method, Chin. Phys. B, 27, 120502.
  26. [26]  Tene, A.G. and Kofane, T.C. (2017), Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative, Chaos, Solitons and Fractals, 98, 88-100.
  27. [27]  Kamdoum,V.T., Sifeu, T.K., Fautso, K.G., Fotsin, H.B., and Talla, K.P. (2018), Coexistence of attractors in autonomous Van der Pol-Duffing jerk oscillator: Analysis, chaos control and synchronization in its fractional-order form, Pramana - J. Phys., 91, 12,
  28. [28]  Matouk, A.E.(2011), Feedback control and synchronization of a fractional-order modified autono mous Van der Pol- Duffing circuit, Commun. Nonlinear Sci. Numer. Simulat, 16, 975-986.
  29. [29]  Nana, B. andWoafo, P. (2006), Synchronization in a ring of four mutually coupled van der Pol oscillators: Theory and experiment, Physical Review E, 74, 1-8.
  30. [30]  Storti, D., and Reinhall, P. (2000), Phase-locked mode stability for coupled van der Pol oscillators, ASME Journal of Vibrations and Acoustics, 122, 318-323.
  31. [31]  Storti, D. and Rand, R. (1982). Dynamics of two strongly coupled van der Pol oscillators, International Journal of Non-Linear Mechanics, 17, 143-152.
  32. [32]  Nascimento, M.A., Varela, H., and Gallas, J.A. (2015), Periodicity hubs and spirals in an electrochemical oscillator, J. Solid State Electrochem, 19, 3287-3296.
  33. [33]  Petras, I. (2011), Fractional-Order Nonlinear System, Higher Education, Press, China.
  34. [34]  Kengne, J., Chedjou., J.C., Kom, M., Kyamakya, K., and Kamdoum, T.V. (2018), Regular oscillations, chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies, Nonlinear Dyn., DOI 10.1007/s11071-013-1195-y.
  35. [35]  Camacho, E., Rand, R., and Howland, H. (2004), Dynamics of two coupled van der Pol oscillators via a bath, Int. J. Solids Struct., 41, 1233-2143.
  36. [36]  Wolf. A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985), Determining Lyapunov Exponents from a Time Series, Physica D, 16, 285-317
  37. [37]  Kengne, R., Tchitnga, R., Mabekou, S., Tekam, R., Soh, G.B., and Fomethe, A. (2018), On the relay coupling of three fractional-order oscillators with time-delay consideration: Global and cluster syn-chronizations, Chaos, Solitons and Fractals, 111, 6-17.
  38. [38]  Cecilia, C., Carlos, A., Briozzo., R.G., Joana, G.F., and Jason, A.C.G. (2013), Periodicity hubs and wide spirals in a two-component autonomous electronic circuit, Chaos, Solitons & Fractals, 52, 59-65.
  39. [39]  Leandro, J. and Jason, A.C.G. (2012), Intricate routes to chaos in the Mackey-Glass delayed feedback system, Physics Letters A, 376, 2109-2116.
  40. [40]  Ott, E., Grebogi, C., and Yorke, J.A. (1990), Controlling chaos, Phys. Rev. Lett. 64, 1196.
  41. [41]  Slutzky, M.W., Cvitanovic, P., and Mogul, D.J. (2003), Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techniques, IEEE Trans. Biomed. Eng., 50, 559.
  42. [42]  Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., and Feliu, V. (2010), Fractional-order systems and controls: fundamentals and applications, Berlin: Springer.
  43. [43]  Viet-Thanh, P., Sifeu, T.K., Christos, V., Sajad, J., and Tomasz, K. (2017), A simple three-dimensional fractional-order chaotic system without equilibrium: Dynamics, circuitry implementation, chaos control and synchronization, Int. J. Electron. Commun. (AEÜ), 78, 220-227.