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

Abstract

In this article, the recently-discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation, as an equation in its own right, with a slowly-varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents to show that solutions which have approached the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous 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 curve in the space-time plane 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. This curve is called the space-time buffer curve. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial time. Hence, a competition arises between these two solutions, as to which one ceases to be exponentially small first, and this competition governs spatial dependence of the DHB. We find four different cases of DHB, depending on the outcomes of the competition, and we quantify to leading order how these depend on the main system parameters, including the Hopf frequency, initial time, initial data, source terms, and diffusivity. Examples are presented for each case, with source terms that are uni-modal, spatially-periodic, smooth step function, and algebraically-growing. 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.

Capturing the cascade: a transseries approach to delayed bifurcations

“…In asymptotic analysis, this corresponds to the crossing of a curve known as an anti-Stokes curve. This suggests that the Stokes phenomenon plays a role in this system behaviour, in a similar fashion to the continuous delayed bifurcations in [38]. In fact, the solution does contain Stokes curves that are responsible for appearance of exponential factors in the solution; however, finding these Stokes curves requires continuing the solution in the negative-n direction, and was therefore not presented here.…”

“…At this stage, it might be expected that we should express the governing equation ( 30) in terms of η, and perform an expansion in this variable; however, a comparison of the terms in (38) and (39) suggests that iterating the map once leads to a simplification. Writing the x dependence explicitly, the equation becomes:…”

“…There has been a significant volume of work studying the asymptotic behaviour of canards in continuous settings, see for example, using composite asymptotic expansions in [10,28,33], steepest descent analysis [38], and Borel summation methods in [25]. Borel summation methods are closely connected to transseries resummation methods (see [2]), and have been used to study discrete problems, as in [54].…”