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.