Two-phase resonant patterns in forced oscillatory systems: boundaries, mechanisms and forms

We use the forced complex Ginzburg–Landau (CGL) equation to study resonance in oscillatory systems periodically forced at approximately twice the natural oscillation frequency. The CGL equation has both resonant spatially uniform solutions and resonant two-phase standing-wave pattern solutions such as stripes or labyrinths. The spatially uniform solutions form a tongue-shaped region in the parameter plane of the forcing amplitude and frequency. But the parameter range of resonant standing-wave patterns does not coincide with the tongue of spatially uniform oscillations. On one side of the tongue the boundary of resonant patterns is inside the tongue and is formed by the nonequilibrium Ising Bloch bifurcation and the instability to traveling waves. On the other side of the tongue the resonant patterns extend outside the tongue forming a parameter region in which standing-wave patterns are resonant but uniform oscillations are not. The standing-wave patterns in that region appear similar to those inside the tongue but the mechanism of their formation is different. The formation mechanism is studied using a weakly nonlinear analysis near a Hopf–Turing bifurcation. The analysis also gives the existence and stability regions of the standing-wave patterns outside the resonant tongue. The analysis is supported by numerical solutions of the forced complex Ginzburg–Landau equation.