论文标题
塔塔恩斯附近的稳态以外的空间定位 - -bogdanov分叉
Spatial localisation beyond steady states in the neighbourhood of the Takens--Bogdanov bifurcation
论文作者
论文摘要
在Takens-Bogdanov（TB）分叉上的干草叉和HOPF分叉的巧合发生在许多物理系统中，例如双排除对流，二元对流和磁反转。在具有周期性边界条件的一个维度上，对相关的正常形式的分析表明，稳定模式，站立波，调制波和行进波的存在，其中术语的系数在正常形式中分类了所有可能的不同的分叉场景中的所有可能的分叉场景（Dangelmayr＆Knobloch，1987年）。在这项工作中，我们基于Swift-Hohenberg方程开发了一种新的简单模式形成的PDE模型，该模型适合于发作时具有结核病正常形式，这使我们能够探索在各种分叉场景中的动态，包括在域中越宽，比图案的长度尺寸较大。我们确定了两个分叉场景，其中不同类型的解决方案之间的共存是正常形式方程的分析。在这些情况下，我们通过检查宽领域中的模式形成来寻找空间局部的解决方案。我们恢复了两种类型的局部状态，即在琐碎状态的背景下的局部稳态和在琐碎状态的背景下的空间局部行驶波的局部稳态状态，这些状态以前在其他系统中已经观察到。此外，我们确定了两种新型的空间定位状态：在调制波背景下的局部稳态和稳态背景下的局部行驶波的局部稳态。 PDE模型很容易在大型域中数值求解，因此将允许在一个或多个维度中进一步研究图案形成，并在一个或多个维度上进行TB分叉，并探索稳态以外的一系列背景和前景模式组合。
The coincidence of a pitchfork and Hopf bifurcation at a Takens-Bogdanov (TB) bifurcation occurs in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in one dimension with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves and travelling waves, where the values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr & Knobloch, 1987). In this work we develop a new and simple pattern-forming PDE model, based on the Swift-Hohenberg equation, adapted to have the TB normal form at onset, which allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localised solutions by examining pattern formation in wide domains. We recover two types of localised states, that of a localised steady state in the background of the trivial state and that of a spatially localised travelling wave in the background of the trivial state which have previously been observed in other systems. Additionally, we identify two new types of spatially localised states: that of a localised steady state in a modulated wave background and that of a localised travelling wave in a steady state background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond steady states.
