Bifurcation to localized oscillations

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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.

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