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Lucas Backes

Publications and source records attributed to Lucas Backes.

36 records · Page 2Linked to original sources
arXiv2020

A general approach to nonautonomous shadowing for nonlinear dynamics

Given a nonautonomous and nonlinear differential equation \begin{equation}\label{DE} x'=A(t)x+f(t,x) \quad t\geq 0, \end{equation} on an arbitrary Banach space $X$, we formulate very general conditions for the associated linear equation $x'=A(t)x$ and for the nonlinear term $f:[0,+\infty)\times X\to X$ under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that $x'=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behaviour as a very particular case. In addition, we require that $f$ is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting a discrete time versions of our results.

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arXiv2020

Hyers-Ulam stability for hyperbolic random dynamics

We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.

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arXiv2020

On the spectral radius of compact operator cocycles

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger-Wang's formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle.

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arXiv2020

Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction.

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arXiv2020

Shadowing for infinite dimensional dynamics and exponential trichotomies

Let $(A_m)_{m\in \Z}$ 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 $m\in \Z$. We prove that whenever the Lipschitz constants of $f_m$, $m\in \Z$, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \Z$, 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.

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arXiv2020

Cohomology of fiber-bunched twisted cocycles over hyperbolic systems

A twisted cocycle taking values on a Lie Group $G$ is a cocycle that, in each step, is twisted by an automorphism of $G$. In the case when $G=GL(d,\mathbb{R})$, we prove that if two Hölder continuous twisted cocycles satisfying the so called fiber-bunching condition have the same periodic data then they are cohomologous.

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arXiv2020

Topological pressure for discontinuous semiflows and a variational principle for impulsive dynamical systems

We introduce four, a priori different, notions of topological pressure for possibly discontinuous semiflows acting on compact metric spaces and observe that they all agree with the classical one when restricted to the continuous setting. Moreover, for a class of \emph{impulsive semiflows}, which are examples of discontinuous systems, we prove a variational principle. As a consequence, we conclude that for this class of systems the four notions coincide and, moreover, they also coincide with the notion of topological pressure introduced in \cite{ACS17}.

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arXiv2018

Periodic approximation of exceptional Lyapunov exponents for semi-invertible operator cocycles

We prove that for semi-invertible and Hölder continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.

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arXiv2018

Shadowing for nonautonomous dynamics

We prove that whenever a sequence of invertible and bounded operators $(A_m)_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential dichotomy and a sequence of differentiable maps $f_m \colon X\to X$, $m\in \mathbb{Z}$, has bounded and Hölder derivatives, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \mathbb{Z}$ 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.

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arXiv2018

A Livšic theorem for matrix cocycles over non-uniformly hyperbolic systems

We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,μ)$ is a non-uniformly hyperbolic system and $A:M \to GL(d,\mathbb{R}) $ is an $α$-Hölder continuous map satisfying $ A(f^{n-1}(p))\ldots A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ and $n\in \mathbb{N}$, there exists a measurable map $P:M\to GL(d,\mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $μ$-almost every $x\in M$. Moreover, we prove that whenever the measure $μ$ has local product structure the transfer map $P$ is $α$-Hölder continuous in sets with arbitrary large measure.

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arXiv2017

A remark on the invertibility of semi-invertible cocycles

We observe that under certain conditions on the Lyapunov exponents a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a Hölder continuous map $A:M\to M(d, \mathbb{R})$ over a hyperbolic map $f:M\to M$ satisfies a Livšic's type condition, that is, if $A(f^{n-1}(p))\cdot\ldots \cdot A(f(p))A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ then the cocycle is invertible, meaning that $A(x)\in GL(d,\mathbb{R})$ for every $x\in M$, and a Livšic's type theorem is satisfied.

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