Abstract
A system of three coupled limit cycle oscillators with vastly different frequencies is studied. The three oscillators, when uncoupled, have the frequencies ω 1=O(1), ω 2=O(1/ε) and ω 3=O(1/ε 2), respectively, where ε≪1. The method of direct partition of motion (DPM) is extended to study the leading order dynamics of the considered autonomous system. It is shown that the limit cycles of oscillators 1 and 2, to leading order, take the form of a Jacobi elliptic function whose amplitude and frequency are modulated as the strength of coupling is varied. The dynamics of the fastest oscillator, to leading order, is unaffected by the coupling to the slower oscillator. It is also found that when the coupling strength between two of the oscillators is larger than a critical bifurcation value, the limit cycle of the slower oscillator disappears. The obtained analytical results are formal and are checked by comparison to solutions from numerical integration of the system.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Blekhman, I.I.: Vibrational Mechanics–Nonlinear Dynamic Effects, General Approach, Application. World Scientific, Singapore (2000)
Belhaq, M., Sah, S.: Horizontal fast excitation in delayed van der Pol oscillator. Commun. Nonlinear Sci. Numer. Simul. 13, 1706–1713 (2008)
Bourkha, R., Belhaq, M.: Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos Solitons Fractals 34(2), 621 (2007)
Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, Berlin (1971)
Coppola, V.T., Rand, R.H.: Averaging using elliptic functions: approximation of limit cycles. Acta Mech. 81, 125–142 (1990)
Fahsi, A., Belhaq, M.: Effect of fast harmonic excitation on frequency-locking in a van der Pol-Mathieu-Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 14(1), 244–253 (2009)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
Jensen, J.S.: Non-trivial effects of fast harmonic excitation. PhD dissertation, DCAMM Report, S83, Dept. Solid Mechanics, Technical University of Denmark (1999)
Nadim, F., Manor, Y., Nusbaum, M.P., Marder, E.: Frequency regulation of a slow rhythm by a fast periodic input. J. Neurosci. 18(13), 5053–5067 (1998)
Barkin, Yu.V., Vilke, V.G.: Celestial mechanics of planet shells. Astron. Astrophys. Trans. 23(6), 533–553 (2004)
Nayfeh, A.H., Chin, C.M.: Nonlinear interactions in a parametrically excited system with widely spaced frequencies. Nonlinear Dyn. 7, 195–216 (1995)
Rand, R.H.: Lecture notes in nonlinear vibrations. Version 52 (2005). http://audiophile.tam.cornell.edu/randdocs
Sah, S., Belhaq, M.: Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator. Chaos Solitons Fractals 37(5), 1489–1496 (2008)
Thomsen, J.J.: Vibrations and Stability, Advanced Theory, Analysis and Tools. Springer, Berlin (2003)
Thomsen, J.J.: Slow high-frequency effects in mechanics: problems, solutions, potentials. Int. J. Bifurc. Chaos 15, 2799–2818 (2005)
Tuwankotta, J.M., Verhulst, F.: Hamiltonian systems with widely separated frequencies. Nonlinearity 16, 689–706 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sheheitli, H., Rand, R.H. Dynamics of three coupled limit cycle oscillators with vastly different frequencies. Nonlinear Dyn 64, 131–145 (2011). https://doi.org/10.1007/s11071-010-9852-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9852-x