ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamics of a System of Two Coupled Oscillators Driven by a Third Oscillator

Journal of Applied Nonlinear Dynamics 3(3) (2014) 271--282 | DOI:10.5890/JAND.2014.09.006

Lauren Lazarus$^{1}$; Richard H. Rand$^{2}$

$^{1}$ Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA

$^{2}$ Department of Mathematics, Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA

Abstract

Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscillators, where each is driven by the same independent third oscillator. The presence of numerous bifurcation curves defines parameter regions with 2, 4, or 6 solutions corresponding to phase locking. In all cases, only one solution is stable. Elsewhere, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m:n.These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.

Acknowledgments

The authors wish to thank Professor Michal Lipson and graduate students Mian Zhang and Shreyas Shah for calling our attention to this problem, which has application to their research.

References

1.  [1] Zhang, M., Wiederhecker, G.S., Manipatruni, S., Barnard, A., McEuen, P., and Lipson, M. (2012), Synchronization of Micromechanical Oscillators Using Light, Physical Review Letters, 109 (23), 233906.
2.  [2] Mendelowitz, L., Verdugo, A., and Rand, R. (2009), Dynamics of three coupled limit cycle oscillators with application to artificial intelligence, Communications in Nonlinear Science and Numerical Simulation, 14 (1), 270–283.
3.  [3] Baesens, C., Guckenheimer, J., Kim, S., MacKay, R.S. (1991), Three coupled oscillators: mode–locking, global bifurcations and toroidal chaos, Physica D, 49 (3), 387–475.
4.  [4] Cohen, A.H., Holmes, P.J., and Rand, R.H. (1982), The Nature of the Coupling Between Segmental Oscillators of the Lamprey Spinal Generator for Locomotion: A Mathematical Model, Journal of Mathematical Biology, 13 (3), 345-369.
5.  [5] Keith, W. L. and Rand, R. H. (1984), 1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators, J. Math. Biology, 20 (2), 133–152.
6.  [6] Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B., Wang, X. (1998), AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations.