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Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator

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Abstract

This paper deals with the analog circuit implementation and synchronization of a model consisting of a van der Pol oscillator coupled to a Duffing oscillator. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of the systems solutions (i.e., elastic coupling). The primary motivation of our investigations lays in the fact that coupled attractors of different types might serve as a good model for real systems in nature (e.g., electromechanical, physical, biological, or economic systems). The stability of fixed points is examined. The bifurcation structures of the system are analyzed with particular emphasis on the effects of nonlinearity. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results shows a very good agreement. By exploiting recent results on adaptive control theory, a controller is designed that enables both synchronization of two unidirectionally coupled systems and the estimation of unknown parameters of the drive system.

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References

  1. Van der Pol, B.: On ‘relaxation-oscillation’. Philos. Mag. Ser. 7 2, 978–992 (1926)

    Google Scholar 

  2. Duffing, G.: Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz and Ihre Technishe Bedentung Friedr. Vieweg & Sohn, Braunschweig (1918)

    Google Scholar 

  3. Han, Y.J.: Dynamics of coupled nonlinear oscillators of different attractors: van der Pol oscillator and damped Duffing oscillator. J. Korean Phys. Soc. 37(1), 3–9 (2000)

    Google Scholar 

  4. Chedjou, J.C., Kyamakya, K., Moussa, I., Kuchenbecker, H.-P., Mathis, W.: Behavior of a self sustained electromechanical transducer and routes to chaos. J. Vib. Acoust. 128, 282–293 (2006)

    Article  Google Scholar 

  5. Hayashi, C.: Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  6. Partliz, C., Lauterborn, W.: Period doubling cascade and devil staircases of the driven van der Pol oscillator. Phys. Rev. A 36, 1428–1434 (1987)

    Article  Google Scholar 

  7. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field. Springer, New York (1983)

    Google Scholar 

  8. Chedjou, J.C., Fotsin, H.B., Woafo, P., Domngang, S.: Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 748–756 (2001)

    Article  MATH  Google Scholar 

  9. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  10. Venkatesan, A., Lakshmanan, M.: Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: numerical and analytical studies. Phys. Rev. E 56, 6321–6330 (1997)

    Article  MathSciNet  Google Scholar 

  11. Kuznetsov, A.P., Stankevich, N.V., Turukina, L.V.: Coupled van der Pol–Duffing oscillators: phase dynamics and structure of synchronization. Physica D 238(14), 1203–1215 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuznetsov, A.P., Roman, J.P.: Synchronization of coupled anisochronous auto-oscillating systems. Nonlinear Phenom. Complex Syst. 12(1), 54–60 (2009)

    MATH  Google Scholar 

  13. Siewe, M.S., Yamgoué, S.B., Moukam Kakmeni, E.M., Tchawoua, C.: Chaos controlling self-sustained electromechanical seismograph system based on Melnikov theory. Nonlinear Dyn. (2010). doi:10.1007/s11071-010-9725-3

    Google Scholar 

  14. Vincent, U.E., Kenfack, A.: Synchronization and bifurcation structures in coupled periodically forced non-identical Duffing oscillator. Phys. Scr. 77, 045005 (2008). doi:10.1088/0031-8949/77/04/045005

    Article  Google Scholar 

  15. Wolf, A., Swift, J.B., Swinney, H.L., Wastano, J.A.: Determining Lyapunov exponents from time series. Physica 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Ozoguz, S., Elwakil, A., Kennedy, M.: Experimental verification of the butterfly attractor in a modified Lorenz system. Int. J. Bifurc. Chaos 12, 1627–1632 (2002)

    Article  Google Scholar 

  17. Li, X.F., Chu, Y.D., Zhang, J.G., Chang, Y.X.: Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor. Chaos Solitons Fractals 41, 2360–2370 (2009)

    Article  MATH  Google Scholar 

  18. Balachandra, V., Kandiban, G.: Experimental and numerical realization of high order autonomous Van der Pol–Duffing oscillator. Indian J. Pure Appl. Phys. 47, 823–827 (2009)

    Google Scholar 

  19. Chen, S., Hu, J., Wang, C., Lü, J.: Adaptive synchronization of uncertain Rössler hyperchaotic system based on parameter identification. Phys. Lett. A 321, 50–55 (2004)

    Article  MATH  Google Scholar 

  20. Chen, D.Y., Wu, C., Liu, C.F., Ma, X.Y., You, Y.J., Zhan, R.F.: Synchronization and circuit simulation of a new double-wing chaos. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-0083-6

    Google Scholar 

  21. Liu, Y.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9960-2

    Google Scholar 

  22. Chen, A., Lu, J., Lü, J., Yu, S.: Generating hyperchaotic Lü attractor via state feedback control. Physica A 364, 103–110 (2006)

    Article  Google Scholar 

  23. Pehlivan, I., Uyaroglu, Y.: A new chaotic attractor from general Lorenz system family and its electronic experimental implementation. Turk. J. Electr. Eng. Comput. Sci. 18(2), 171–184 (2010)

    Google Scholar 

  24. Rulkov, N.F.: Images of synchronized chaos: experiments with circuits. Chaos 6(3), 262–279 (1996)

    Article  Google Scholar 

  25. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoes analysis and circuit implementation for a four-wing chaotic attractor. Nonlinear Dyn. 65(1–2), 131–140 (2011)

    Article  MATH  Google Scholar 

  26. Itoh, M.: Synthesis of electronic circuits for simulating nonlinear dynamics. Int. J. Bifurc. Chaos 11(3), 605–653 (2001)

    Article  Google Scholar 

  27. Tsay, S.C., Huang, C.K., Qiu, D.L., Chen, W.T.: Implementation of bidirectional chaotic communication systems based on Lorenz circuits. Chaos Solitons Fractals 20, 567–579 (2004)

    Article  MATH  Google Scholar 

  28. Yujun, N., Xingyuan, W., Mingjun, W., Huaguang, Z.: A new hyperchaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 15, 3518–3524 (2010)

    Article  Google Scholar 

  29. Buscarino, A., Fortuna, L., Frasca, M.: Experimental robust synchronization of hyperchaotic circuits. Physica D 238, 1917–1922 (2010)

    Article  Google Scholar 

  30. Hamil, D.C.: Learning about chaotic circuits with SPICE. IEEE Trans. Ed. 36(1), 28–35 (1991)

    Google Scholar 

  31. Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  32. Vincent, U.E.: Chaos synchronization using active control and backstepping control: a comparative analysis. Nonlinear Anal. Model. Control 13(2), 253–261 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Callenbach, L., Linz, S.J., Hanggi, P.: Synchronization in simple chaotic flows. Phys. Lett. A 287, 90–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pikovsky, A.S., Rosenblum, M.G., Kurth, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  35. Fotsin, H.B., Moukam Kakmeni, F.M., Bowong, S.: An adaptive observer for chaos synchronization of a nonlinear electronic circuit. Int. J. Bifurc. Chaos 16(9), 2671–2679 (2006)

    Article  MATH  Google Scholar 

  36. Fotsin, H.B., Woafo, P.: Adaptive synchronization of uncertain chaotic Van der Pol–Duffing oscillator based on parameter identification. Chaos Solitons Fractals 24, 1363–1371 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, Y., Zhou, W., Fang, J., Sun, W.: Adaptive bidirectionally coupled synchronization of chaotic systems with unknown parameters. Nonlinear Dyn. (2010). doi:10.1007/s11071-010-9911-3

    Google Scholar 

  38. Adloo, H., Roopi, M.: Review article on adaptive synchronization of chaotic systems with unknown parameters. Nonlinear Dyn. (2010). doi:10.1007/s11071-010-9880-6

    Google Scholar 

  39. Liao, T.L., Tsai, S.H.: Adaptive synchronization of chaotic systems and its application to secure communications. Chaos Solitons Fractals 11, 1387–1396 (2000)

    Article  MATH  Google Scholar 

  40. Ming, L., Jing, J.: A new theorem to synchronization of unified chaotic systems via adaptive control. J. Univ. Sci. Technol. Beijing 10, 72–76 (2003)

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Automatique et d’Informatique Appliquée (LAIA), IUT-FV Bandjoun, University of Dschang, BP 134, Bandjoun, Cameroon

    J. Kengne, G. Kenne & G. H. Kom

  2. Institute of Smart Systems Technologies, University of Klagenfurt, Universitaetsstr. 65, 9020, Klagenfurt, Austria

    J. C. Chedjou & K. Kyamakya

Authors
  1. J. Kengne
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  2. J. C. Chedjou
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  3. G. Kenne
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  4. K. Kyamakya
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  5. G. H. Kom
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Correspondence to J. Kengne.

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Kengne, J., Chedjou, J.C., Kenne, G. et al. Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator. Nonlinear Dyn 70, 2163–2173 (2012). https://doi.org/10.1007/s11071-012-0607-8

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  • DOI: https://doi.org/10.1007/s11071-012-0607-8

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