Delayed Hopf Bifurcation and Space–Time Buffer Curves in the Complex Ginzburg–Landau Equation
Delayed Hopf Bifurcation and Space–Time Buffer Curves in the Complex Ginzburg–Landau Equation
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复Ginzburg-Landau方程中的延迟Hopf分岔和时空缓冲曲线
DOI：
10.1093/imamat/hxac001
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发表时间：
2022
影响因子：
1.2
通讯作者：
Vo, Theodore
中科院分区：
文献类型：
--
作者：
Goh, Ryan;Kaper, Tasso J;Vo, Theodore
本文分析了反应-扩散偏微分方程（PDEs）中最近发现的延迟Hopf分岔（DHB）现象，并将三次复Ginzburg-Landau方程作为一个参数缓慢变化的方程进行了分析。我们首先利用经典的定相和最陡下降的渐近方法对线性化PDE进行分析，证明了在Hopf分岔之前接近吸引准稳态（QSS）的解在瞬时Hopf分岔和QSS变得排斥后仍在该状态附近保持很长时间。在复时间平面上，线性化PDE的相函数有一个鞍点，Stokes线和反Stokes线是渐近的中心。非线性项采用迭代法处理由扰动引起的线性特解的偏微分方程的温和形式。这跟踪了在完整的非线性PDE中，吸引和排斥QSS附近解的接近度。接下来，我们证明了在穿过鞍形的关键Stokes线之外，在时空平面上有一条曲线，沿着这条曲线，线性PDE的特解不再是指数小，导致非线性PDE的解偏离排斥性QSS并表现出大振幅振荡。这条曲线叫做时空缓冲曲线。齐次解也不再以空间依赖的方式呈指数小，这也是由初始数据和时间决定的。因此，这两种解决方案之间产生了竞争，即哪一种解决方案首先不再是指数小的，这种竞争支配着DHB的空间依赖性。根据竞争的结果，我们发现了四种不同的DHB情况，我们量化了这些情况对主要系统参数的依赖程度，包括Hopf频率、初始时间、初始数据、源项和扩散率。给出了每种情况下的例子，源项是单峰函数、光滑阶跃函数、空间周期函数和代数增长函数。此外，在dhb后振荡中还观察到丰富的时空动态。最后，研究表明，可以设计大振幅源项，使解决方案在排斥QSS附近花费更长的时间，因此，可以实现对振荡延迟发作的区域特定控制。
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-linear 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.