Tzou J C, Ma Y-P, Bayliss A, Matkowsky B J, Volpert V A
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Feb;87(2):022908. doi: 10.1103/PhysRevE.87.022908. Epub 2013 Feb 14.
Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic Hopf frequency. Such solutions of this nonvariational problem are in contrast to the stationary pinning solutions found in the subcritical Turing regime for the variational Swift-Hohenberg equations, characterized by a spatially periodic pattern embedded in a spatially homogeneous background state. Numerical continuation was used to solve periodic boundary value problems in time for the Fourier amplitudes of the spatiotemporal Turing-Hopf pinning solutions. The solution branches are organized in a series of saddle-node bifurcations similar to the known snaking structures of stationary pinning solutions. We find two intertwined pairs of such branches, one with a defect in the middle of the striped region, and one without. Solutions on one branch of one pair differ from those on the other branch by a π phase shift in the spatially periodic region, i.e., locations of local minima of solutions on one branch correspond to locations of maxima of solutions on the other branch. These branches are connected to branches exhibiting collapsed snaking behavior, where the snaking region collapses to almost a single value in the bifurcation parameter. Solutions along various parts of the branches are described in detail. Time dependent depinning dynamics outside the saddle nodes are illustrated, and a time scale for the depinning transitions is numerically established. Wavelength variation within the snaking region is discussed, and reasons for the variation are given in the context of amplitude equations. Finally, we compare the pinning region to the Maxwell line found numerically by time evolving the amplitude equations.
研究了一维布鲁塞尔振子模型在余维二图灵 - 霍普夫点附近的时空图灵 - 霍普夫钉扎解。图灵分岔和霍普夫分岔均为超临界且稳定的。钉扎解表现出临界波长附近的静止条纹与特征霍普夫频率附近的时间周期振荡共存。这种非变分问题的解与变分斯威夫特 - 霍恩伯格方程在亚临界图灵区域中发现的静止钉扎解形成对比，后者的特征是在空间均匀背景态中嵌入空间周期性图案。利用数值延拓法求解时空图灵 - 霍普夫钉扎解的傅里叶振幅随时间的周期边值问题。解分支以一系列鞍结分岔的形式组织起来，类似于已知的静止钉扎解的蛇行结构。我们发现了两对相互交织的此类分支，一对在条纹区域中间有缺陷，另一对没有。一对分支中的一个分支上的解与另一个分支上的解在空间周期区域中相差π相移，即一个分支上解的局部最小值位置对应于另一个分支上解的最大值位置。这些分支与表现出塌缩蛇行行为的分支相连，在这种情况下，蛇行区域在分岔参数中塌缩到几乎一个单一值。详细描述了沿分支各部分的解。说明了鞍结外与时间相关的解钉扎动力学，并通过数值确定了解钉扎转变的时间尺度。讨论了蛇行区域内的波长变化，并在振幅方程的背景下给出了变化的原因。最后，我们将钉扎区域与通过对振幅方程进行时间演化数值得到的麦克斯韦线进行了比较。
