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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.

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For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy [AFK11]. From spectral theory side, the "Schrödinger conjecture" [AFK11] and the "Last's intersection spectrum conjecture" have been verified [JM12a]. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian [FK09, Kri] and Jitomirskaya-Marx [JM12a, MJ17]. In this paper, we prove the above mentioned results for ultra-differentiable potentials.1 We refer to Section 2.1 for the definitions and basic results. 2 Here α is Diophantine (denote α ∈ DC(v, τ )), if there exist v > 0 and τ > 1 such thatWe also denote DC := v>0, τ >1 DC(v, τ ) the union.