Stability of Nonautonomous Differential Equations

The classical notion of exponential dichotomy, which we call uniform exponential dichotomy, is a considerable restriction for the dynamics and it is important to look for more general types of hyperbolic behavior. These generalized notions can be much more typical than the notion of uniform exponential dichotomy. This is precisely the case of the notion of nonuniform exponential dichotomy, that we introduce in Section 2.1. In particular, we show that essentially any (nonautonomous) linear differential equation admits such a dichotomy (see Section 2.3). We note that each uniform exponential dichotomy is also a nonuniform exponential dichotomy.

We give in this chapter conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a linear equation v´ = A(t)v persists under suf- ficiently small linear perturbations. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and the “angles” between the stable and unstable subspaces. The proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of certain contractions. We emphasize that we do not need the notion of admissibility (with respect to bounded nonlinear perturbations). We also establish related results in the case of strong nonuniform exponential dichotomies. All the results are obtained in Banach spaces. The presentation follows closely [18].

We want to construct stable and unstable invariant manifolds without assuming the existence of a uniform exponential dichotomy for the linear variational equation. Our main objective is to describe the weakest possible setting under which one can construct the invariant manifolds.We still require some amount of hyperbolicity. Namely, we show that under fairly general assumptions the generalized notion of nonuniform exponential dichotomy allows us to establish the existence of stable and unstable invariant manifolds. In this chapter we only consider “Lipschitz manifolds”, that is, graphs of Lipschitz functions. We refer to Chapters 5 and 6 for the existence of smooth invariant manifolds (respectively in Rn and in arbitrary Banach spaces), under slightly stronger assumptions. We follow closely [12], although now considering the general case when the stable and unstable subspaces may depend on the time t. Lipschitz center manifolds were obtained with a similar approach in [8]; we refer to Chapter 8 for the construction of smooth center manifolds.

In this chapter we start the study of the regularity of the Lipschitz manifolds constructed in Chapter 4. We only consider stable manifolds. As in Section 4.5, the theory for unstable manifolds is analogous, and the proofs can be readily obtained by reversing the time. We only consider in this chapter the case of finite-dimensional spaces. This is due to the method of proof of the smoothness of the invariant manifolds, which uses in a decisive manner the compactness of the closed unit ball in Rn (in the proof of Lemma 5.11). The proof is based on the construction of an invariant family of cones, in a similar manner to that in the classical hyperbolic theory, although now using an appropriate family of Lyapunov norms. The family of cones allows us to obtain an invariant distribution which coincides with the tangent bundle of the invariant manifold. This also allows us to discuss the continuity of the distribution, and thus the continuity of the tangent spaces, that corresponds to the smoothness of the invariant manifold. We note that we deal directly with the semiflows instead of first considering time-1 maps as it is sometimes customary in hyperbolic dynamics. The infinite-dimensional case is treated in Chapter 6 with an entirely different approach, although at the expense of requiring more regularity for the vector field. The material in this chapter is taken from [6] (for Sections 5.1–5.4) and [5] (for Sections 5.5–5.6), although now considering the general case when the stable and unstable subspaces may depend on the time t.

We establish in this chapter the existence of smooth stable manifolds for semiflows defined by nonautonomous differential equations in a Banach space. One can obtain unstable manifolds simply by reversing the time. We also establish the exponential decay on the stable manifold of the derivatives of the semiflow with respect to the initial condition (see (6.8) and (6.9)). We are not aware of any similar result in the literature even in the case of uniform exponential dichotomies. Our approach to the proof of the stable manifold theorem consists again in using the differential equation and the invariance of the stable manifold under the dynamics to conclude that it must be the graph of a function satisfying a certain fixed point problem.

A fundamental problem in the study of the local behavior of a dynamical system is whether the linearization of the system along a given solution approximates well the solution itself in some open neighborhood. In other words, we look for an appropriate local change of variables, called a conjugacy, that can transform the system into a linear one. Moreover, as a means to distinguish the dynamics in a neighborhood of the solution further than in the topological category (such as, for example, to distinguish different types of nodes), the change of variables should be as regular as possible. The problem goes back to the pioneering work of Poincaré, that can be interpreted today as looking for an analytic change of variables which transforms the initial system into a linear one. The work of Sternberg [89, 90] showed that there are algebraic obstructions, expressed in terms of resonances between the eigenvalues of the linear approximation, that prevent the existence of conjugacies with a prescribed high regularity (see also [19, 20, 87, 61] for further related work). The main purpose of this chapter is to establish a nonautonomous and nonuniform version of the Grobman–Hartman theorem in Banach spaces. In addition, we show that the conjugacies are always Hölder continuous, with Hölder exponent expressed in terms of ratios of Lyapunov exponents. We follow closely [17, 10].

Center manifold theorems are powerful tools in the analysis of the behavior of dynamical systems. For example, when the equation v´ = A(t)v has a (uniformly) partially hyperbolic behavior with no unstable directions, then under some mild additional assumptions all solutions of v´ = A(t)v + f(t, v) converge exponentially to the center manifold. Hence, the stability of the system is completely determined by the behavior on the center manifold. Therefore, one often considers a reduction to the center manifold. This has also the advantage of reducing the dimension of the system. Furthermore, since one often needs to approximate the center manifolds to sufficiently high order, it is also important to discuss their regularity. Our main goal is to establish the existence of smooth invariant center manifolds in the presence of nonuniformly partially hyperbolic behavior. The method of proof is inspired in the arguments of Chapter 6. In particular, the smoothness of the center manifolds is obtained with a single fixed point problem, instead of one for each additional derivative. We follow closely [15], now with arbitrary stable and unstable subspaces.

We show in this chapter that for a nonautonomous differential equation (in the presence of a nonuniform exponential trichotomy; see Chapter 8), the (time) reversibility and equivariance of the associated semiflow descends respectively to the reversibility and equivariance in any center manifold. We note that time-reversal symmetries are among the fundamental symmetries in many “physical” systems, both in classical and quantum mechanics. This is due to the fact that many Hamiltonian systems are reversible (see [53] for many examples). In spite of the crucial differences between reversible and equivariant dynamical systems, the techniques that are useful in any of the two contexts usually carry over to the other one. This will be apparent along the exposition. We follow closely [14].

We show in this chapter that any linear equation v´ = A(t)v, with A(t) in block form with blocks corresponding to the stable and center-unstable components, admits a strong nonuniform exponential dichotomy. While the extra exponentials in the notion of nonuniform exponential dichotomy substantially complicate the study of invariant manifolds in former chapters, we are able to obtain fairly general results at the expense of a careful control of the nonuniformity. In particular, we showed that if the equation v´ = A(t)v has a nonuniform exponential dichotomy with sufficiently small nonuniformity (when compared to the Lyapunov exponents), then with mild assumptions on the perturbation f there exist stable and unstable manifolds for the nonlinear equation v´ = A(t)v+f(t, v). We note that we do not need the nonuniformity to be zero, only sufficiently small. Therefore, it is important to estimate in quantitative terms how much a nonuniform exponential dichotomy can deviate from a uniform one. Fortunately, there exists a device, introduced by Lyapunov, that allows one to measure this deviation. It is the so-called notion of regularity (see Section 10.1 for the definition), introduced by Lyapunov in his doctoral thesis [57] (the expression is his own), which nowadays seems unfortunately apparently overlooked in the theory of differential equations. We emphasize that we only consider finite-dimensional spaces in this chapter. The infinite-dimensional case is considered in Chapter 11. The material in this chapter is based in [13], which in its turn is inspired in [1]. See [16] for a related study in the case of the discrete time.