Oscillons in the planar Ginzburg-Landau equation with 2: 1 forcing

Oscillons are spatially localized, time-periodic structures that have been observed in many natural processes, often under temporally periodic forcing. Near Hopf bifurcations, such systems can be formally reduced to forced complex Ginzburg-Landau equations, with oscillons then corresponding to stationary localized patterns. In this manuscript, stationary localized structures of the planar 2 : 1 forced Ginzburg-Landau equation are investigated analytically and numerically. The existence of these patterns is proved in regions where two spatial eigenvalues collide at zero. A numerical study complements these analytical results away from onset.