We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.Secondly, we construct the center manifold for skew systems driven by the external forcing.Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

中文翻译：

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PDE中随时间变化的中心歧管

我们考虑演化形式的外力方程。从数学上讲，这些是由有限维动力系统驱动的偏斜系统。我们治疗中包含的两个非常常见的情况是准周期强迫和随机过程强迫。我们允许演化是一个PDE，即使它没有适当的位置，也没有定义一个流程（并非所有初始条件都导致一个解）。我们首先建立一个通用的抽象定理，在适当的条件下（光谱，非变性，平滑度等）假设，确定了“时间相关不变流形”（TDIM）的存在。这些歧管随着强迫而发展。它们使得原始方程始终与歧管中的矢量场相切。因此，对于TDIM中的初始数据，原始方程等效于一个常微分方程。这使我们能够通过研究有限维系统的解来定义完整方程的解族。请注意，即使原始方程式不适当地，并且不接受任意初始条件的解（TDIM选择存在解的初始条件），该策略也可能适用。其次，我们构造了由外部强迫驱动的偏斜系统的中心流形。第三，我们给出了抽象结果在微分方程中的具体应用，这些微分方程的线性算子是拟三角函数式，且受准方程约束。周期性的扰动。TDIM的使用使我们能够建立准周期解的存在并研究共振的影响。