DCDS
Admissibility versus nonuniform exponential behavior for noninvertible cocycles
Luis Barreira Claudia Valls
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
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
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
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
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
The Boussinesq system:dynamics on the center manifold
Claudia Valls
In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Boussinesq system given by

$ \partial_t u = -\partial_x v - \alpha \partial_{x x x}v - \partial_x(u v), \quad \partial_t v = - \partial_x u - v \partial_x v,$

For $\alpha \leq 1$, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every $\alpha$, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.

keywords: Boussinesq system Hamiltonian formalism energy modes
CPAA
Centers for polynomial vector fields of arbitrary degree
Jaume Llibre Claudia Valls
We present two new families of polynomial differential systems of arbitrary degree with centers, a two--parameter family and a four--parameter family.
keywords: polynomial vector fields arbitrary degree Centers
CPAA
Topological conjugacies and behavior at infinity
Luis Barreira Claudia Valls
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
CPAA
Center manifolds for nonuniform trichotomies and arbitrary growth rates
Luis Barreira Claudia Valls
We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.
keywords: Center manifolds nonuniform exponential trichotomies.
CPAA
Stability of some waves in the Boussinesq system
Claudia Valls
In this paper we study analytically a class of waves in the variant of the classical Boussinesq system given by

$\partial_t u = -\partial_x v - \alpha \partial_{x x x} v - \epsilon \partial_x(u v), \quad \partial_t v = - \partial_x u - \epsilon v \partial_x v,$

where $\epsilon$ is an small parameter and $\alpha \in (0,1)$. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for some values of $\alpha$, it contains solutions that are defined for large values of time and they are very close (of order $O(\epsilon)$) to a linear torus for long times (of order $O(\epsilon^{-1})$). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

keywords: normal forms Hamiltonian formalism. Boussinesq system

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