Phase bistability and phase bistable patterns in self-oscillatory systems under a resonant periodic forcing with spatially modulated amplitude

I consider the problem of self-oscillatory systems undergoing a homogeneous Hopf bifurcation when they are submitted to an external forcing that is periodic in time, at a frequency close to the systems natural frequency (1:1 resonance), and whose amplitude is slowly modulated in space. Starting from a general, unspecified model and making use of standard multiple scales analysis, I show that the close-to-threshold dynamics of such systems is universally governed by a generalized, complex Ginzburg-Landau (CGL) equation. The nature of the generalization depends on the strength and of other features of forcing: (i) For generic, sufficiently weak forcings the CGL equation contains an extra, inhomogeneous term proportional to the complex amplitude of forcing, as in the usual 1:1 resonance with spatially uniform forcing; (ii) For stronger perturbations, whose amplitude sign alternates across the system, the CGL equation contains a term proportional to the complex conjugate of the oscillations envelope, like in the classical 2:1 resonance, responsible for the emergence of phase bistability and of phase bistable patterns in the system. Finally I show that case (ii) is retrieved from case (i) in the appropriate limit so that the latter can be regarded as the generic model for the close-to-threshold dynamics of the type of systems considered here. The kind of forcing studied in this work thus represents an alternative to the classical parametric forcing at twice the natural frequency of oscillations and opens the way to new forms of pattern formation control in self-oscillatory systems, what is especially relevant in the case of systems that are quite insensitive to parametric forcing, such as lasers and other nonlinear optical cavities.