Spatially localized oscillations in periodically forced systems are intriguing phenomena. They may occur in spatially homogeneous media (oscillons), but quite often emerge in heterogeneous media, such as the auditory system, where localized oscillations are believed to play an important role in frequency discrimination of incoming sound waves. In this paper, we use an amplitude-equation approach to study the spatial profile of the oscillations and the factors that affect it. More specifically, we use a variant of the forced complex Ginzburg-Landau (FCGL) equation to describes an oscillatory system below the Hopf bifurcation with space-dependent Hopf frequency, subject to both parametric and additive forcing. We show that spatial heterogeneity, combined with bistability of system states, results in spatial asymmetry of the localized oscillations. We further identify parameters that control that asymmetry, and characterize the spatial profile of the oscillations in terms of maximum amplitude, location, width and asymmetry. Our results bear qualitative similarities to empirical observation trends that have found in the auditory system.
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