Smoothness Of Invariant Manifolds Research Articles

On invariant manifolds and invariant foliations without a spectral gap

It is known that a spectral gap condition is required for the existence and smoothness of invariant manifolds and invariant foliations near a fixed point. However, any planar mapping with a unipotent linear part, having a center manifold on the whole phase plane, does not have such a gap. In this paper, we give the existence and smoothness of invariant manifolds, which are invariant submanifolds of the center manifold, for the aforementioned planar mapping. Corresponding to the invariant submanifolds, we further prove the existence and smoothness of invariant foliations on the center manifold.

Stability in Nonautonomous Dynamics: A Survey of Recent Results

We discuss recent results in the stability theory of nonautonomousdifferential equations under sufficiently small perturbations.We mostly concentrate on the description of our own work, with emphasison the Lyapunov stability of solutions, the existence and smoothnessof invariant manifolds, and the construction and regularity of topologicalconjugacies, among other topics. The main novelty is that we alwaysconsider a nonuniform exponential behavior of the linear variationalequations, given either by the existence of a nonuniform exponentialcontraction or a nonuniform exponential dichotomy.

Stability in Nonautonomous Dynamics: A Survey of Recent Results

We discuss recent results in the stability theory of nonautonomousdifferential equations under sufficiently small perturbations.We mostly concentrate on the description of our own work, with emphasison the Lyapunov stability of solutions, the existence and smoothnessof invariant manifolds, and the construction and regularity of topologicalconjugacies, among other topics. The main novelty is that we alwaysconsider a nonuniform exponential behavior of the linear variationalequations, given either by the existence of a nonuniform exponentialcontraction or a nonuniform exponential dichotomy.

Invariant Manifolds for Abstract Functional Differential Equations and Related Volterra Difference Equations in a Banach Space

For abstract functional differential equations (FDE) and Volterra difference equations (VDE) in a Banach space, the local existence and smoothness of invariant manifolds, such as stable/unstable manifolds, center-stable/center-unstable manifolds and center manifolds, are established by means of the variation of constants formula in the phase space in [18] and [12]. Also, it is shown that in a neighborhood of the zero solution the behavior of solutions of FDE (resp. VDE) is described, in some sense, by a certain ordinary differential equation (resp. first order difference equation) in a finite dimensional space. As a corollaly, the principle of linearized stability is derived.

Lyapunov-type numbers and torus breakdown: numerical aspects and a case study

In this work we examine the breakdown mechanism for a 2-torus associated to a system of coupled oscillators. To this end, we rely on computation of the Lyapunov-type numbers, as defined by Fenichel in N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971) 193–226. We give some general results on these numbers, and some specific constructive results for the case in which there is phase-locking on the torus. This is the situation for the system of coupled oscillators we consider, and it leads to great simplifications in the computation of the Lyapunov-type numbers.

On proving existence and smoothness of invariant manifolds in singular perturbation problem

On proving existence and smoothness of invariant manifolds in singular perturbation problem

An abstract approach to smoothness of invariant manifolds

We establish a general abstract Cm-fixed-point theorem for a finite scale of Banach spaces, extending to the Cm-case a previous C1-fixed-point theorem of Vanderbauwhede and Van Gils. This result gives short proofs of Cm-smoothness for center-, center-unstable, inertial and other center-like invariant manifolds for various types of differential and difference equations in finite and infinite dimensions. It is also applicable to fixed-point equations on LP-spaces. In the present paper we present applications to invariant manifolds for evolution equations with time-dependent nonlinearities.

Smoothness of invariant manifolds

In this paper we present some functional analytic tools that allow us to prove a theorem on C k,η -smoothness of general (discrete and continuous) invariant manifolds for C k,η -systems. A crucial role in this proof will be played by a theorem on C k,η -parameter dependence of a fixed point of a contraction defined on a scale of Banach spaces. This fixed point principle is stated independently and may be of interest in itself. Further functional analytic considerations concerning smoothness properties of the Nemytski operator are presented in a rather general form. Throughout the paper we treat the two cases of continuous and discrete dynamical systems in a unified manner. Thereby we use some concepts and results from the so-called calculus on measure chains which was developed by the author in Res. in Math. 18, 1990, 18–56. Knowledge of this calculus is not necessary for understanding the paper.

The Construction and Smoothness of Invariant Manifolds by the Deformation Method

This paper proves optimal results for the invariant manifold theorems near a fixed point for a mapping (or a differential equation) by using the deformation, or Lie transform, method from singularity theory. The method was inspired by the difficulties encountered by the implicit function theorem technique in the case of the center manifold. The idea here is simply to deform the given system into its linearization and to track this deformation using the flow of a time-dependent vector field. Corresponding to the difficulties with the center manifold encountered by other techniques, we run into a “derivative loss” in this case as well, which is overcome by utilizing estimates on the differentiated equation. A survey of the other methods used in the literature is also presented.