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I consider the problem of self-oscillatory systems undergoing a homogeneousHopf bifurcation when they are submitted to an external forcing that isperiodic in time, at a frequency close to the system's natural frequency (1:1resonance), and whose amplitude is slowly modulated in space. Starting from ageneral, unspecified model and making use of standard multiple scales analysis,I show that the close-to-threshold dynamics of such systems is universallygoverned by a generalized, complex Ginzburg-Landau (CGL) equation. The natureof 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 theusual 1:1 resonance with spatially uniform forcing; (ii) For strongerperturbations, whose amplitude sign alternates across the system, the CGLequation contains a term proportional to the complex conjugate of theoscillations envelope, like in the classical 2:1 resonance, responsible for theemergence of phase bistability and of phase bistable patterns in the system.Finally I show that case (ii) is retrieved from case (i) in the appropriatelimit so that the latter can be regarded as the generic model for theclose-to-threshold dynamics of the type of systems considered here. The kind offorcing studied in this work thus represents an alternative to the classicalparametric forcing at twice the natural frequency of oscillations and opens theway to new forms of pattern formation control in self-oscillatory systems, whatis especially relevant in the case of systems that are quite insensitive toparametric forcing, such as lasers and other nonlinear optical cavities.

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