Hartman Theorem Research Articles

A generalized Grobman–Hartman theorem for nonautonomous dynamics

The purpose of this note is to extend the recent generalized version of the Grobman–Hartman theorem established by Bernardes Jr. and Messaoudi from an autonomous to nonautonomous dynamics. More precisely, we prove that any sufficiently small perturbation of a nonautonomous linear dynamics that admits a generalized exponential dichotomy is topologically conjugated to its linear part. In addition, we prove that under certain mild additional conditions, the conjugacy is in fact Hölder continuous.

Smooth linearization under nonuniform hyperbolicity

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

C1 Hartman Theorem for random dynamical systems

The C1 Hartman Theorem states that every C1,1 contraction in Rd admits C1 linearization. In this paper we prove C1,β linearizability for C1,α contractive (or expansive) random diffeomorphisms with appropriate constants 0<α<1 and 0<β<1, which implies a random version of the C1 Hartman Theorem. Moreover, the conjugacy is proved to be tempered. This result strengthens related results even in the deterministic case, lowering the C1,1 condition to C1,α and enhancing the C1 smoothness of the conjugacy to C1,β in Rd. In the proof we use a smooth weak-stable invariant manifold in the random sense to construct the conjugacy and to overcome difficulties from nonuniformity and measurability.

Shadowing for infinite dimensional dynamics and exponential trichotomies

Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $. We prove that whenever the Lipschitz constants of $f_m$, $m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in {\mathop Z} $, has various types of shadowing. Moreover, if X is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers–Ulam stability for certain difference equations and we obtain a very general version of the Grobman–Hartman's theorem for nonautonomous dynamics.

Perturbations of delay equations

We study the behavior of an exponential dichotomy for a nonautonomous linear delay equation under sufficiently small linear and nonlinear perturbations. In particular, we establish a partial version of the Grobman–Hartman theorem and we use the characterization of an exponential dichotomy in terms of an admissibility property to establish the robustness property.

New Results on Linearization of Differential Equations with Piecewise Constant Argument

In this work, we give a version of Grobman–Hartman theorem for the nonautonomous differential equations with piecewise constant argument of generalized type when the nonlinear term is unbounded and its linear system partially satisfies $$\alpha $$-exponential dichotomy. More specifically, we divide the linear system into three subsystems, one of the subsystems is not necessary to admit $$\alpha $$-exponential dichotomy. It was assumed that the whole linear system admits exponential dichotomy in Pinto and Robledo (Z Anal Anwend 37:101–126, 2018), Zou and Shi (J Appl Anal Comput 7(1):309–333, 2017) and Zou et al. (Qual Theory Dyn Syst 18:495–531, 2019). Thus, we extend and improve the previous results in the literature. Moreover, we prove that the conjugated function H(t, x) is also $$\omega $$-periodic when the systems are $$\omega $$-periodic.

Shadowing for Nonautonomous Dynamics

Abstract We prove that whenever a sequence of bounded operators ( A m ) m ∈ ℤ {(A_{m})_{m\in\mathbb{Z}}} acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps f m : X → X {f_{m}\colon X\to X} , m ∈ ℤ {m\in\mathbb{Z}} , has bounded and Hölder derivatives, the nonautonomous dynamics given by x m + 1 = A m ⁢ x m + f m ⁢ ( x m ) {x_{m+1}=A_{m}x_{m}+f_{m}(x_{m})} , m ∈ ℤ {m\in\mathbb{Z}} , has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.

A Grobman–Hartman Theorem for Differential Equations with Piecewise Constant Arguments of Mixed Type

We obtain sufficient conditions for the existence of a uniformly and Hölder continuous homeomorphism between the solutions of a linear differential system with piecewise constant argument of generalized type and the solutions of a perturbed family. The main tool is a recently introduced definition of exponential dichotomy.

Hölder Regularity of Grobman–Hartman Theorem for Dynamic Equations on Measure Chains

It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Holder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Holder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Holder continuous.

Tempered exponential dichotomies: admissibility and stability under perturbations

ABSTRACTWe give a characterization of the notion of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. We note that for a linear cocycle over a measure-preserving transformation satisfying a certain integrability assumption, it follows from the multiplicative ergodic theorem that the dynamics admits a tempered exponential dichotomy if and only if all Lyapunov exponents are non-zero almost everywhere. As a consequence of our approach, we give a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations and we establish a version of the Grobman–Hartman theorem yielding the existence of topological conjugacies between a linear dynamics with a tempered exponential dichotomy and any sufficiently small nonlinear perturbation. In addition, we show that the conjugacy maps vary continuously with the perturbation.

Dynamical equivalence of impulsive quasilinear equations

Abstract Using Green type map we can find sufficient conditions under which an impulsive quasilinear equation is dynamically equivalent to its corresponding linear equation. This result extends Grobman Hartman theorem for equations without ordinary dichotomy.

Stability of L1 contractions

AbstractThe notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

DYNAMICAL EQUIVALENCE OF QUASILINEAR EQUATIONS

Using Green type map we can find sufficient conditions under which a quasilinear equation is dynamically equivalent to its corresponding linear equation. This result extends Grobman - Hartman theorem for equations without ordinary dichotomy.

Topological conjugacies and behavior at infinity

We obtain a version of the Grobman--Hartman theorem in Banachspaces for perturbations of a nonuniform exponential contraction, both for discrete and continuous time. More precisely, we consider the general case of an exponential contraction with an arbitrary nonuniform part and obtained from a nonautonomous dynamics, and we establish the existence of Hölder continuous conjugacies between an exponential contraction and any sufficiently small perturbation. As a nontrivial application, we describe the asymptotic behavior of the topological conjugacies in terms of the perturbations: namely, we show that for perturbations growing in a certain controlled manner the conjugacies approach zero at infinity and that when the perturbations decay exponentially at infinity the conjugacies have the same exponential behavior.

Conjugacies for impulsive equations

AbstractFor impulsive differential equations, we construct topological conjugacies between linear and nonlinear perturbations of non-uniform exponential dichotomies. In the case of linear perturbations, the topological conjugacies are constructed in a more or less explicit manner. In the nonlinear case, we obtain an appropriate version of the Grobman–Hartman Theorem for impulsive equations, with a simple and direct proof that involves no discretization of the dynamics.

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.

Reversibility of Conjugacies for Nonautonomous Equations

We establish a reversible version of the Grobman–Hartman theorem for nonautonomous differential equations in Banach spaces. Namely, we show that if the differential equation is reversible, then the topological conjugacies in the Grobman–Hartman theorem also commute with the symmetries. The result is illustrated with an example.

A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows

We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.

On the topological equivalence of some fuzzy flows near hyperbolic equilibria

The purpose of this work is to study fuzzy dynamical systems associated with deterministic systems. The Grobman–Hartman theorem states that, near hyperbolic equilibria, there exists a homeomorphism between the nonlinear system's trajectories and the linearized correspondent system's trajectories. That is, these systems are topologically equivalent. A theorem similar to Grobman–Hartman theorem to fuzzy flows is the main result in this article. For fuzzy flows obtained from each system it states that the nonlinear and the linearized are topologically equivalent.

A Grobman–Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators A m , we establish a version of the Grobman–Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates e c ρ ( n ) , determined by a sequence ρ ( n ) . For all sufficiently small Lipschitz perturbations A m + f m we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators A m . We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ ( n ) = n , but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ ( n ) = log n .