arXiv subjects

Davor Dragičević

Publications and source records attributed to Davor Dragičević.

At least 19 recordsLinked to original sources

arXiv2026

Regularity of the variance in quenched CLT for random intermittent dynamical systems

We study random dynamical systems composed of LSV maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025), e70150).

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arXiv2024

Takens Theorem for nonautonomous partially hyperbolic dynamical systems

Takens Theorem for a partially hyperbolic dynamics provides a normal linearization along the center manifold. In this paper, we give the nonautonomous version of Takens Theorem under non-resonance conditions formulated in terms of the dichotomy spectrum. In our proof, one difficulty is to solve homological equations for the normal form theory which involve a center variable, while another difficulty is to find the dichotomy spectrum of a certain matrix cocycle that is block lower triangular. In order to overcome those difficulties, in comparison with the autonomous case, we need an additional term in the (nonautonomous) non-resonance conditions to guarantee certain spectral gap conditions. This additional term disappears naturally in the autonomous case.

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arXiv2024

Hyers-Ulam stability for finite-dimensional nonautonomous dynamics

The main purpose of this paper is to obtain necessary and sufficient conditions under which a nonautonomous, finite-dimensional and two-sided dynamics generated by a sequence of matrices or a linear ODE exhibits Hyers-Ulam stability. Specifically, in the case of discrete time we consider a nonautonomous difference equation with possibly noninvertible coefficients, while in the case of continuous time we deal with a nonautonomous ordinary differential equation without any bounded growth assumptions.

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arXiv2023

Parameterized shadowing for nonautonomous dynamics

For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit $C^k$, $k\in \N$ shadowing with respect to a parameter. Our results are applicable to situations when the linear part is not hyperbolic. In the case when the linear part is hyperbolic, we obtain results dealing with parameterized Hyers-Ulam stability.

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arXiv2023

Smooth Linearization of Nonautonomous Coupled Systems

In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n, y_n), \quad y_{n+1}=g_n( y_n), \end{equation*} onto the solutions of the associated partially linearized uncoupled system \begin{equation*} x_{n+1} =A_nx_n, \quad y_{n+1}=g_n( y_n). \end{equation*} In the present work we go one step further and provide conditions under which $H_n$ and $H_n^{-1}$ are smooth in one of the variables $x$ and $y$. We emphasise that our conditions are of a general form and do not involve any kind of dichotomy, nonresonance or spectral gap assumptions for the linear part which are present on most of the related works.

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arXiv2023

A vector-valued almost sure invariance principle for random expanding on average cocycles

We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel's approach for establishing ASIP and the recent construction of the so-called adapted norms, which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of a scaling condition, answering a question of the first and third authors.

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arXiv2022

Quenched linear response for smooth expanding on average cocycles

We establish an abstract quenched linear response result for random dynamical systems, which we then apply to the case of smooth expanding on average cocycles on the unit circle. In sharp contrast to the existing results in the literature, we deal with the class of random dynamics that does not necessarily exhibit uniform decay of correlations. Our techniques rely on the infinite-dimensional ergodic theory and in particular, on the study of the top Oseledets space of a parametrized transfer operator cocycle.

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arXiv2022

Description of tempered exponential dichotomies by admissibility with no Lyapunov norms

Tempered exponential dichotomy formulates the nonuniform hyperbolicity for random dynamical systems. It was described by admissibility of a pair of function classes defined with Lyapunov norms, For MET-systems (systems satisfying the assumptions of multiplicative ergodic theorem (abbreviated as MET)), it can be described by admissibility of a pair without a Lyapunov norm. However, it is not known how to choose a suitable Lyapunov norms before a tempered exponential dichotomy is given, and there are examples of random systems which are not MET-systems but have a tempered exponential dichotomy. In this paper we give a description of tempered exponential dichotomy for general random systems, which may not be MET-systems, purely by measurable admissibility of three pairs of function classes with no Lyapunov norms. Further, restricting to the MET-systems, we obtain a simpler description of only one pair with no Lyapunov norms. Finally, we use our results to prove the roughness of tempered exponential dichotomies for parametric random systems and give a Hölder continuous dependence of the associated projections on the parameter.

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arXiv2022

On the linearization of infinite-dimensional random dynamical systems

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds for infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the non-linear perturbations, we don't require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we don't prescribe any behaviour for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.

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arXiv2022

Multiscale linearization of nonautonomous systems

We present sufficient conditions under which a given linear nonautonomous system and its nonlinear perturbation are topologically conjugated. Our conditions are of a very general form and provided that the nonlinear perturbations are well-behaved, we do not assume any asymptotic behaviour of the linear system. Moreover, the control on the nonlinear perturbations may differ along finitely many mutually complementary directions. We consider both the cases of one-sided discrete and continuous dynamics.

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arXiv2021

Quenched limit theorems for expanding on average cocycles

We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for non-autonomous dynamics developed by Dragičević et al., to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory. Our results provide an affirmative answer to a question posed by Buzzi.

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arXiv2021

Statistical stability and linear response for random hyperbolic dynamics

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis rely on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

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arXiv2021

Almost sure invariance principle for random dynamical systems via Gouëzel 's approach

We extend the spectral approach of S. Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behavior of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.

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arXiv2021

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 the 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.

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arXiv2021

Linearization and H\" older Continuity for Nonautonomous Systems

We consider a nonautonomous system \[ \dot x=A(t)x+f(t,x,y),\quad \dot y = g(t,y)\] and give conditions under which there is a transformation of the form $H(t,x,y)=(x+h(t,x,y),y)$ taking its solutions onto the solutions of the partially linearized system \[ \dot x=A(t)x,\quad \dot y = g(t,y).\] Shi and Xiong \cite{SX} proved a special case where $g(t,y)$ was a linear function of $y$ and $\dot x=A(t)x$ had an exponential dichotomy. Our assumptions on $A$ and $f$ are of the general form considered by Reinfelds and Steinberga \cite{RS}, which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove H\" older continuity of $H$ and its inverse in $x$ and $y$. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when $\dot x=A(t)x$ is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results.

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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

Limit theorems for random expanding or hyperbolic dynamical systems and vector-valued observables

The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for \emph{vector-valued} observables. We stress that the previous works considered exclusively the case of scalar-valued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviation principle, concentration inequalities, Berry-Esseen estimates as well as Edgeworth and large deviation expansions. Although our techniques rely on the approach developed in the previous works of the first author \textit{et al}., we emphasize that our arguments require several nontrivial adjustments as well as new ideas.

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arXiv2018

Optimal linear responses for Markov chains and stochastically perturbed dynamical systems

The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the little-studied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.

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