Classification of Spatially Localized Oscillations in Periodically Forced Dissipative Systems

Formation of spatially localized oscillations in parametrically driven systems is studied, focusing on the dominant 2: 1 resonance tongue. Both damped and self-excited oscillatory media are considered. Near the primary subharmonic instability such systems are described by the forced complex Ginzburg-Landau equation. The technique of spatial dynamics is used to identify three basic types of coherent states described by this equation-small amplitude oscillons, large amplitude reciprocal oscillons resembling holes in an oscillating background, and fronts connecting two spatially homogeneous states oscillating out of phase. In many cases all three solution types are found in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The origin of this behavior can be traced to the formation of a heteroclinic cycle in space between the finite amplitude spatially homogeneous phase-locked oscillation and the zero state. The results provide an almost complete classification of the properties of spatially localized states within the one-dimensional forced complex Ginzburg-Landau equation as a function of the coefficients.