Robustness via Lyapunov functions

For nonautonomous linear equations x^′=A(t)x, we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic Lyapunov functions. Nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies. In particular, we construct explicitly strict Lyapunov functions for each exponential dichotomy. As a nontrivial application, we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. This represents a considerable simplification of former work.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

Stability in delay difference equations with nonuniform exponential behavior

We study the stability under perturbations for delay difference equations in Banach spaces. Namely, we establish the (nonuniform) stability of linear nonuniform exponential contractions under sufficiently small perturbations. We also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy, and show that the stable manifolds are Lipschitz in the perturbation.     

Lyapunov Functions for General Nonuniform Dichotomies

For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.     

Lyapunov sequences for exponential dichotomies

For a linear nonautonomous dynamics with discrete time, we study the relation between nonuniform exponential dichotomies and strict Lyapunov sequences. Given such a sequence, we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence. The main difficulty is to extract some information about the angles between the stable and unstable subspaces (or some appropriate notion in the case of Banach spaces) from the Lyapunov sequence. In particular, for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic Lyapunov sequences, that is, strict Lyapunov sequences defined by quadratic forms. We also construct explicitly families of strict Lyapunov sequences for each nonuniform exponential dichotomy, in terms of Lyapunov norms.     

Nonlinear perturbations of nonuniform exponential dichotomy on measure chains

This paper focuses on nonlinear perturbations of flows in Banach spaces, corresponding to a nonautonomous dynamical system on measure chains admitting a nonuniform exponential dichotomy. We first define the nonuniform exponential dichotomy of linear nonuniformly hyperbolic systems on measure chains, then establish a new version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics on measure chains with the help of nonuniform exponential dichotomies. Moreover, we also construct stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy. In particular, it is shown that the stable invariant manifolds are Lipschitz in the initial values provided that the nonlinear perturbation is a sufficiently small Lipschitz perturbation.     

Nonuniform exponential contractions and Lyapunov sequences

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict Lyapunov sequences. We note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. We also obtain “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. We also show how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. Moreover, we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.     

Lyapunov regularity of impulsive differential equations

For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.     

Robustness of nonuniform exponential dichotomies in Banach spaces

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x^′=A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the “angles” between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v^′=A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v^′=A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.     

Nonuniform Dichotomies via Lyapunov Functions

For a nonautonomous linear equation x′ =  A(t)x we show how to characterize a nonuniform exponential dichotomy using strict Lyapunov functions. In particular, the stable and unstable subspaces are obtained from invariant families of cones determined by each Lyapunov function. We also obtain converse theorems, constructing explicitly a family of strict Lyapunov functions for each nonuniform exponential dichotomy. We emphasize that nonuniform exponential dichotomies include as a very particular case (uniform) exponential dichotomies.     

Lyapunov functions for trichotomies with growth rates

We consider linear equations x^′=A(t)x that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates ecρ(t) determined by a function ρ(t). For example, the usual exponential behavior with ρ(t)=t is included as a very special case, and when ρ(t)=logt we obtain a polynomial behavior. We emphasize that we also consider the general case of nonuniform exponential behavior, which corresponds to the existence of what we call a ρ-nonuniform exponential trichotomy. This is known to occur in a large class of nonautonomous linear equations. Our main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a ρ-nonuniform exponential trichotomy. This includes criteria for the existence of a ρ-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions we show that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomies in terms of strict Lyapunov functions, we establish the robustness of ρ-nonuniform exponential trichotomies under sufficiently small linear perturbations. We emphasize that in comparison with former works, our proof of the robustness is much simpler even when ρ(t)=t.     

Admissibility on the half line for evolution families

For an arbitrary evolution family, we consider the notion of an exponential dichotomy with respect to a family of norms and characterize it completely in terms of the admissibility of bounded solutions, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a nonuniform exponential dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of an exponential dichotomy with respect to a family of norms under sufficiently small Lipschitz and C ¹ parameterized perturbations. Moreover, we establish the optimal regularity of the dependence on the parameter of the projections onto the stable spaces of the perturbation.     

Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.     

A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics

We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps Am+fm such that the linear parts Am admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps Am+fm and Am. Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations fm: we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.     

Robustness of nonuniform behavior for discrete dynamics

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we establish the robustness of a nonuniform dichotomy, in the sense that the existence of such a dichotomy for a given linear cocycle persists under sufficiently small linear perturbations. The robustness result for the nonuniform contractions is also established.     

Robustness for impulsive equations

For nonautonomous linear impulsive differential equations in Banach spaces, we establish the robustness of exponential contractions and exponential dichotomies, in the sense that the exponential behavior persists under sufficiently small linear perturbations. We also consider the more general case of nonuniform exponential behavior.     

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C¹-parameterized perturbations originate a family of exponential dichotomies of class C¹ in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form eλρ(t) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ(t)=t.