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. (paper)