Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

Abstract

In this article, the phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation with a slowly-varying parameter. We use the classical asymptotic methods of stationary phase and steepest descents to show that solutions which approach the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The nonlinear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS. Next, we show that beyond a key Stokes line through the saddle there is a space-time buffer curve along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the nonlinear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial time. We find four different cases of DHB, depending on the competition between the homogeneous and particular solutions, and we quantify how these depend on system parameters. Examples are presented for each case, with uni-modal, spatially-periodic, smooth step, and algebraically-growing source terms. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence region-specific control over the delayed onset of oscillations can be achieved.