Hopf分岔附近的图灵不稳定性和模式
Turing Instabilities and Patterns Near a Hopf Bifurcation
摘要：
本文给出了具有Neumann边界条件的两组分反应扩散方程组发生图灵不稳定性的一个充分必要条件。我们将这个条件应用到反应扩散系统中，该系统由具有一个不动点和一个超临界Hopf分支的向量场构成。对于非线性扩散方程和Ginzburg-Landau反应扩散方程，我们得到了时间周期解和渐近稳定解之间的转移的分支图（Turing模式）.在两组分反应扩散方程组中，证明了图灵不稳定性的存在性对于图灵模式型解的存在性既不是必要的，也不是充分的.图灵模式可以存在于与局部向量场相关联的Hopf分支的两侧，并且根据初始条件，时间周期解和稳定解可以共存。
英文摘要：
We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated to the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.
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