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Bifurcation to localized oscillations

Published online by Cambridge University Press:  01 August 1997

RACHEL KUSKE  and
THOMAS ERNEUX
RACHEL KUSKE
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
THOMAS ERNEUX
Affiliation:
Optique nonlinéaire théorique, Université Libre de Bruxelles, Campus Plaine CP 231,B-1050 Bruxelles, Belgium
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Abstract

A singular perturbation method is applied to a system of two weakly coupled strongly non-linear but non-identical oscillators. For certain parameter regimes, stable localized solutions exist for which the amplitude of one oscillator is an order of magnitude smaller than the other. The leading-order dynamics of the localized states is described by a new system of coupled equations for the phase difference and scaled amplitudes. The degree and stability of the localization has a non-trivial dependence on coupling strength, detuning, and the bifurcation parameter. Three distinct types of localized behaviour are obtained as solutions to these equations, corresponding to phase-locking, phase-drift, and phase-entrainment. Quantitative results for the phases and amplitudes of the oscillators and the stability of these phenomena are expressed in terms of the parameters of the model.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 8 , Issue 4 , August 1997 , pp. 389 - 402
Copyright
© 1997 Cambridge University Press

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