图灵鸭解
Les Canards de Turing
AI总结：
本文研究扩散系数差异极大的反应扩散方程，发现图灵分岔附近涌现的空间周期鸭解族，并用几何奇异摄动理论证明其存在性，揭示可逆折叠奇点与尺度不变性在其中的关键作用。
摘要：
本文研究一类扩散系数相差悬殊的反应扩散方程组。我们报告了从奇异图灵分岔中涌现出的空间周期鸭解族的发现。这些空间周期鸭解渐近地接近于图灵分岔而出现，而图灵分岔在空间常微分方程系统中是可逆的1:1共振Hopf分岔，这一现象在空间动力学中类似于时间相关常微分方程中Hopf分岔附近鸭爆炸里极限环鸭解的出现。在整个偏微分方程系统中，我们证明在所研究的大部分参数值下，图灵分岔是亚临界的，并给出了一些直接数值模拟的结果，表明若干不同类型的空间鸭模式是原型偏微分方程中的吸引子。为支持这些数值发现，我们运用几何去奇异化和几何奇异摄动理论来证明这些空间周期鸭解族的存在性。关键的是，在奇异极限下，我们研究了一类新的可逆折叠奇点。特别地，存在两个II型可逆折叠鞍结分岔（RFSN-II），每一个都渐近地出现在图灵分岔附近。我们推导出这些奇点的解析公式，并证明它们的鸭解在所观测到的空间周期鸭解族中起着关键作用。然后，对于分岔参数取值区间中进一步低于图灵分岔和RFSN-II点的部分，空间常微分方程也具有空间周期鸭模式，但这些模式是由可逆折叠鞍点（而非RFSN-II）产生的。研究还发现存在一种有趣的尺度不变性，使得某些空间鸭解的某些分量表现出近乎自相似的动力学。
英文摘要：
In this article, we study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {\em singular Turing bifurcations}. The emergence of these spatially periodic canards asymptotically close to the Turing bifurcations, which are reversible 1:1 resonant Hopf bifurcations in the spatial ODE system, is an analog in spatial dynamics of the emergence of limit cycle canards in the canard explosions that occur asymptotically close to Hopf bifurcations in time-dependent ODEs. In the full PDE system, we show that for most parameter values under study the Turing bifurcation is sub-critical, and we present the results of some direct numerical simulations showing that several of the different types of spatial canard patterns are attractors in the prototypical PDE. To support the numerical discoveries, we use geometric desingularization and geometric singular perturbation theory to demonstrate the existence of these families of spatially periodic canards. Crucially, in the singular limit, we study a novel class of {\em reversible folded singularities}. In particular, there are two reversible folded saddle-node bifurcations of type II (RFSN-II), each occurring asymptotically close to a Turing bifurcation. We derive analytical formulas for these singularities and show that their canards play key roles in the observed families of spatially periodic canard solutions. Then, for an interval of values of the bifurcation parameter further below the Turing bifurcation and RFSN-II point, the spatial ODE also has spatially periodic canard patterns, however these are created by a reversible folded saddle (instead of the RFSN-II). It also turns out that there is an interesting scale invariance, so that some components of some spatial canards exhibit nearly self-similar dynamics.
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