SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5703-5747, January 2020.
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.

中文翻译：

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时空鸭和延迟分叉的局部理论

SIAM数学分析杂志，第52卷，第6期，第5703-5747页，2020年1月。
我们提出了一个严格的框架，用于对带有时间标度分离的广泛的无限维动力系统中的鸭嘴和通过分叉的缓慢通道进行局部分析。该框架适用于将快速变量的无穷维动力学系统与缓慢变量的有限维动力学系统耦合的模型。我们证明了这种类型的泛型模型的中心流形的存在，并研究了层问题中（可能）构图稳态的分叉附近的减小的有限维动力学。理论结果辅以详细的示例和数值模拟，包括局部和非局部反应扩散方程，神经场模型和时滞微分方程系统。我们为最近在文献中报道的数值观测提供了分析基础，例如时空鸭囊和在参数缓慢变化的空间扩展系统中穿过Hopf分叉的缓慢通道。我们还提供了局部和非局部模型中通过图灵分支的慢速通道的理论分析。