The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos. 1. Introduction In recent years, considerable research efforts had been devoted to the analysis of coupled limit-cycle oscillators models due to the useful insight they provide into the collective dynamics of various physical, biological, chemical, and electrical (e.g., coupled van der Pol-Duffing electrical circuit) systems [1–5]. The system of two coupled van der Pol oscillators is one of the canonical models exhibiting the mutual synchronisation behaviour [6]. Coupled self-oscillators demonstrate some classical effects such as phase locking of the oscillations with different ratios of the frequencies [7]. Amplitude death or quenching is possible in case of dissipative coupling [8, 9]. This phenomenon has been observed experimentally in the system of coupled electromechanical oscillators [10], coupled electronic systems [11], and coupled biological oscillators [11] just to name a few. Multistability, chaos, and nonisochronism are some special effects of the synchronization picture presented by researchers [12–17]. Introducing additional nonlinearity of Duffing type in the original van der Pol equation leads to the so-called van der Pol-Duffing equation for which potential contains a component in the form of the coordinate raised to the fourth power [18–20]. This
References
[1]
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, UK, 2001.
[2]
J. M. Gonzalez-Miranda, Synchronization and Control of Chaos: An Introduction for Scientists and Engineers, Imperial College Press, London, UK, 2004.
[3]
B. van der Pol, “On relaxation-oscillations,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Sciences Series, vol. 7, no. 2, pp. 978–992, 1926.
[4]
R. Yamapi and P. Woafo, “Dynamics and synchronization of coupled self-sustained electromechanical devices,” Journal of Sound and Vibration, vol. 285, no. 4-5, pp. 1151–1170, 2005.
[5]
H. B. Fotsin and P. Woafo, “Adaptive synchronization of a modified and uncertain chaotic van der Pol-Duffing oscillator based on parameter identification,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1363–1371, 2005.
[6]
A. Balanov, N. Janson, D. Postnov, and O. Sosnovtseva, Synchronization: From Simple to Complex, Springer, Berlin, Germany, 2009.
[7]
M. Belhaq and A. Fahsi, “Frequency-locking in nonlinear forced oscillators near 3:1 and 4:1 resonances,” Annals of Solid and Structural Mechanics, vol. 4, no. 1-2, pp. 15–23, 2012.
[8]
A. P. Kuznetsov, E. P. Seleznev, and N. V. Stankevich, “Nonautonomous dynamics of coupled van der Pol oscillators in the regime of amplitude death,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3740–3746, 2012.
[9]
Y. P. Emelianova, A. P. Kuznetsov, I. R. Sataev, and L. V. Turukina, “Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators,” Physica D: Nonlinear Phenomena, vol. 244, no. 1, pp. 36–49, 2013.
[10]
Y. Zhai, I. Z. Kiss, and J. L. Hudson, “Amplitude death through a hopf bifurcation in coupled electrochemical oscillators: experiments and simulations,” Physical Review E, vol. 69, no. 2, Article ID 026208, 2004.
[11]
I. Ozden, S. Venkataramani, M. A. Long, B. W. Connors, and A. V. Nurmikko, “Strong coupling of nonlinear electronic and biological oscillators: reaching the “amplitude death” regime,” Physical Review Letters, vol. 93, no. 15, Article ID 158102, 2004.
[12]
D. W. Storti and R. H. Rand, “Dynamics of two strongly coupled van der Pol oscillators,” International Journal of Non-Linear Mechanics, vol. 17, no. 3, pp. 143–152, 1982.
[13]
T. Chakraborty and R. H. Rand, “The transition from phase locking to drift in a system of two weakly coupled van der pol oscillators,” International Journal of Non-Linear Mechanics, vol. 23, no. 5-6, pp. 369–376, 1988.
[14]
I. Pastor, V. M. Pérez-García, F. Encinas, and J. M. Guerra, “Ordered and chaotic behavior of two coupled van der Pol oscillators,” Physical Review E, vol. 48, no. 1, pp. 171–182, 1993.
[15]
M. Poliashenko, S. R. McKay, and C. W. Smith, “Hysteresis of synchronous-asynchronous regimes in a system of two coupled oscillators,” Physical Review A, vol. 43, no. 10, pp. 5638–5641, 1991.
[16]
M. Poliashenko, S. R. McKay, and C. W. Smith, “Chaos and nonisochronism in weakly coupled nonlinear oscillators,” Physical Review A, vol. 44, no. 6, pp. 3452–3456, 1991.
[17]
I. Pastor-Díaz and A. López-Fraguas, “Dynamics of two coupled van der Pol oscillators,” Physical Review E, vol. 52, no. 2, pp. 1480–1489, 1995.
[18]
A. P. Kuznetsov and J. P. Roman, “Synchronization of coupled anisochronous auto-oscillating systems,” Nonlinear Phenomena in Complex Systems, vol. 12, no. 1, pp. 54–60, 2009.
[19]
A. P. Kuznetsov and J. P. Roman, “Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization,” Physica D: Nonlinear Phenomena, vol. 238, no. 16, pp. 1499–1506, 2009.
[20]
A. P. Kuznetsov, N. V. Stankevich, and L. V. Turukina, “Coupled van der Pol-Duffing oscillators: phase dynamics and structure of synchronization tongues,” Physica D: Nonlinear Phenomena, vol. 238, no. 14, pp. 1203–1215, 2009.
[21]
J.-W. Ryu, D.-S. Lee, Y.-J. Park, and C.-M. Kim, “Oscillation quenching in coupled different oscillators,” Journal of the Korean Physical Society, vol. 55, no. 2, pp. 395–399, 2009.
[22]
C. A. K. Kwuimy and P. Woafo, “Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm,” Nonlinear Dynamics, vol. 53, no. 3, pp. 201–213, 2008.
[23]
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field, Springer, New York, NY, USA, 1983.
[24]
A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985.
[25]
C. A. Kitio Kwuimy and P. Woafo, “Experimental realization and simulation of a self-sustained macroelectromechanical system,” Mechanics Research Communications, vol. 37, no. 1, pp. 106–110, 2010.
[26]
J. C. Chedjou, H. B. Fotsin, P. Woafo, and S. Domngang, “Analog simulation of the dynamics of a van der pol oscillator coupled to a duffing oscillator,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 6, pp. 748–757, 2001.
[27]
J. Kengne, J. C. Chedjou, G. Kenne, K. Kyamakya, and G. H. Kom, “Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator,” Nonlinear Dynamics, vol. 70, no. 3, pp. 2163–2173, 2012.
