Abstract
The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors.
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Pastor, L., Perez-Garcia, V. M., Encinas-Sanz, F., and Guerra, J. M., ‘Ordered and chaotic behavior of two coupled van der Pol oscillators,’ Physical Review E 48, 1993, 171–182.
Stagliano, J. J., Wersinger, J. M., and Slaminka, E. E., ‘Doubling bifurcations of destroyed T 2 tori,’ Physica D 92, 1996, 164 –177.
Koziowski, J., Parlitz, U., and Lauterborn, W., ‘Bifurcation analysis of two coupled periodically driven Duffing oscillator,’ Physical Review E 51, 1995, 1861–1867.
Paul, R. S. and Rajasekar, S., ‘Migration control in two coupled Duffing oscillators,’ Physical Review E 55, 1997, 6237–6240.
Paul, R. S., Rajasekar, S., and Murali, K., ‘Coexisting chaotic attractors, their basin of attractions and synchronization of chaos in two coupled Duffing oscillators,’ Physics Letters A 264, 1999, 283–288.
Yin, H. W. and Dai, J. H., ‘Phase effect of two coupled periodically driven Duffing oscillators,’ Physical Review E 58, 1998, 5683–5688.
Attilio, M., ‘Parametric excitation for two internally resonant van der Pol oscillators,’ Nonlinear Dynamics 27, 2002, 367–383.
Rajasekar, S. and Murali, K., ‘Resonance behaviour and jump phenomenon in two coupled Duffing-van der Pol oscillators,’ Chaos, Solitons and Fractals 19, 2004, 925–934.
Moukam-Kakmeni, F. M., Bowong, S., Tchawoua, C., and Kaptouom, E., ‘Dynamics and chaos control in nonlinear electrostatic transducers,’ Chaos, Solitons and Fractals 21, 2004, 1093–1108.
Chen, Y. S., The Theory of Nonlinear Dynamics and Bifurcation, Higher Education Press, Beijing, 1993.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Kenfack, A., ‘Bifurcation structure of two coupled periodically driven double-well Duffing oscillators,’ Chaos, Solitons and Fractals 15, 2003, 205–218.
Thompson, J. M. T., Stewart, H. B., and Ueda,Y., ‘Safe, explosive, and dangerous bifurcations in dissipative dynamical systems,’ Physical Review E 49, 1994, 1019–1027.
Osipov, G., Glatz, L., and Troger, H., ‘Suppressing chaos in the Duffing oscillator by impulsive actions,’ Chaos, Solitons and Fractals 9, 1998, 307–321.
Litvak-Hinenzon, A. and Rom-Kedar, V., ‘Symmetry-breaking perturbations and strange attractors,’ Physical Review E 55, 1997, 4964–4978.
Kim, S. Y. and Kim, Y., ‘Dynamic stabilization in the double-well Duffing oscillator,’ Physical Review E 61, 2000, 6517–6520.
Pecora, L. and Carroll, T. L., ‘Pseudoperidic driving: Eliminating multiple domains of attraction using chaos,’ Physical Review Letters 67, 1991, 945–949.
Chacón, R. and García-Hoz, A. M., ‘Bifurcations and chaos in a parametrically damped two-well Duffing oscillator subjected to symmetric periodic pulses,’ Physical Review E 59, 1999, 6558–6568.
Wolf, A., Swift, J., Swinney, H., and Vastano, A., ‘Determining Lyapunov exponents from a time series,’ Physica D 16, 1985, 285–317.
Zheng, Z. G., Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems, Higher Education Press, Beijing, 2004.
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Wei, X., Ruihong, L. & Shuang, L. Resonance and Bifurcation in a Nonlinear Duffing System with Cubic Coupled Terms. Nonlinear Dyn 46, 211–221 (2006). https://doi.org/10.1007/s11071-006-9024-1
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DOI: https://doi.org/10.1007/s11071-006-9024-1