DCDS
Hölder Grobman-Hartman linearization
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2007, 18(1): 187-197 doi: 10.3934/dcds.2007.18.187
We prove that the conjugacies in the Grobman-Hartman theorem are always Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. We also consider the case of hyperbolic trajectories of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. All the results are obtained in Banach spaces. It is common knowledge that some authors claimed that the Hölder regularity of the conjugacies is well known, however without providing any reference. In fact, to the best of our knowledge, the proof appears nowhere in the published literature.
keywords: Grobman--Hartman theorem Hölder regularity.
DCDS
Stability of nonautonomous equations and Lyapunov functions
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2631-2650 doi: 10.3934/dcds.2013.33.2631
We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting nonuniform exponential contractions in terms of strict Lyapunov functions. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniform exponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works, the persistence of the asymptotic stability under sufficiently small linear and nonlinear perturbations.
keywords: nonuniform exponential contractions. Lyapunov functions
DCDS
Regularity of center manifolds under nonuniform hyperbolicity
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2011, 30(1): 55-76 doi: 10.3934/dcds.2011.30.55
We construct $C^k$ invariant center manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a nonuniform exponential trichotomy. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. In addition, we can also consider linear perturbations with the same method.
keywords: center manifolds. Nonuniform exponential dichotomies
DCDS
Stable manifolds with optimal regularity for difference equations
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1537-1555 doi: 10.3934/dcds.2012.32.1537
We obtain stable invariant manifolds with optimal $C^k$ regularity for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.
keywords: Difference equations stable manifolds.
DCDS
Admissibility versus nonuniform exponential behavior for noninvertible cocycles
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1297-1311 doi: 10.3934/dcds.2013.33.1297
We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.
keywords: nonuniform exponential dichotomies. Admissibility
DCDS-S
Delay equations and nonuniform exponential stability
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - S 2008, 1(2): 219-223 doi: 10.3934/dcdss.2008.1.219
For nonautonomous linear delay equations $v'=L(t)v_t$ admitting a nonuniform exponential contraction, we establish the nonuniform exponential stability of the equation $v'=L(t) v_t +f(t,v_t)$ for a large class of nonlinear perturbations.
keywords: nonuniform exponential stability. Delay equations
DCDS
Growth rates and nonuniform hyperbolicity
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2008, 22(3): 509-528 doi: 10.3934/dcds.2008.22.509
We consider linear equations $v'=A(t)v$ that may exhibit different asymptotic behaviors in different directions. These can be thought of as stable, unstable and central behaviors, although here with respect to arbitrary asymptotic rates $e^{c \rho(t)}$ determined by a function $\rho(t)$, including the usual exponential behavior $\rho(t)=t$ as a very special case. In particular, we consider the notion of $\rho$-nonuniform exponential trichotomy, that combines simultaneously the nonuniformly hyperbolic behavior with arbitrary asymptotic rates. We show that for $\rho$ in a large class of rate functions, any linear equation in block form in a finite-dimensional space, with three blocks having asymptotic rates $e^{c \rho(t)}$ respectively with $c$ negative, zero, and positive, admits a $\rho$-nonuniform exponential trichotomy. We also give explicit examples that cannot be made uniform and for which one cannot take $\rho(t)=t$ without making all Lyapunov exponents infinite. Furthermore, we obtain sharp bounds for the constants that determine the exponential trichotomy. These are expressed in terms of appropriate Lyapunov exponents that measure the growth rate with respect to the function $\rho$.
keywords: Asymptotic behavior growth rates Lyapunov exponents.
DCDS
Reversibility and equivariance in center manifolds of nonautonomous dynamics
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2007, 18(4): 677-699 doi: 10.3934/dcds.2007.18.677
We consider reversible and equivariant dynamical systems in Banach spaces, either defined by maps or flows. We show that for a reversible (respectively, equivariant) system, the dynamics on any center manifold in a certain class of graphs (namely $C^1$ graphs with Lipschitz first derivative) is also reversible (respectively, equivariant). We consider the general case of center manifolds for a nonuniformly partially hyperbolic dynamics, corresponding to the existence of a nonuniform exponential trichotomy of the linear variational equation. We also consider the case of nonautonomous dynamics.
keywords: nonuniform exponential trichotomies center manifolds reversibility. equivariance
DCDS
Noninvertible cocycles: Robustness of exponential dichotomies
Luis Barreira Claudia Valls
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4111-4131 doi: 10.3934/dcds.2012.32.4111
For the dynamics defined by a sequence of bounded linear operators in a Banach space, we establish the robustness of the notion of exponential dichotomy. This means that an exponential dichotomy persists under sufficiently small linear perturbations. We consider the general cases of a nonuniform exponential dichotomy, which requires much less than a uniform exponential dichotomy, and of a noninvertible dynamics or, more precisely, of a dynamics that may not be invertible in the stable direction.
keywords: Exponential dichotomies linear perturbations noninvertible dynamics robustness. nonuniform exponential dichotomies
CPAA
Topological conjugacies and behavior at infinity
Luis Barreira Claudia Valls
Communications on Pure & Applied Analysis 2014, 13(2): 687-701 doi: 10.3934/cpaa.2014.13.687
We obtain a version of the Grobman--Hartman theorem in Banach spaces 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.
keywords: topological conjugacies. Exponential contractions

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