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  1. Abstract In this article, the recently discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction–diffusion partial differential equations (PDEs) is analysed 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 on the linearized PDE 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 linearized PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The non-linear 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 in the full, non-linear PDE. 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 non-linearmore »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 data and 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 a uni-modal function, a smooth step function, a spatially periodic function and an algebraically growing function. 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.« less
    Free, publicly-accessible full text available March 2, 2023
  2. Abstract We consider pattern-forming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearisation about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilises with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectral stability of the front in L 2 ( R ) . The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivise the linear flow, and use Riemann surface unfolding in combination with a superpositionmore »principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati–Evans function, and can be located using winding number and parity arguments.« less