Skew-product Flows Research Articles

The weak Smale horseshoe and mean hyperbolicity

The paper concerns the mean hyperbolicity of the skew-product flows induced by the perturbation equations driven by varieties of non-periodic forcings. The weakly averaged horseshoe can be constructed as a mean hyperbolic invariant set in the Poincare section for high dimensional phase space. Due to the non-periodic property, the Poincare return map restricted to the weakly averaged horseshoe region can semi-conjugate to the full Bernoulli shift on infinite symbols, which implies the infinitely many independent choices on the length of return time. As a direct application of mean hyperbolicity, we extend the shadowing lemma due to Liao to the general nonautonomous dynamical systems.

On Uniform Exponential Trisplitting of Discrete Skew-product Semiflows

In the present paper the concept of uniform exponential trisplitting for skew-product flows in Banach space is considered. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. Connections between these concepts are presented and some illustrating examples prove that these are distinct. Also, we present necessary and sufficient conditions for the uniform exponential trisplitting concept with invariant and strongly invariant projectors. Finally, a characterisation in terms of Lyapunov sequences is given.

Exponential Decay of Correlations for Nonuniformly Hyperbolic Flows with a $${{C^{1+\alpha}}}$$ C 1 + α Stable Foliation, Including the Classical Lorenz Attractor

We prove exponential decay of correlations for a class of $C^{1+\alpha}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.

A splitting theorem for affine skew‐product flows

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy.

Admissibility and exponential trichotomy of dynamical systems described by skew-product flows

The aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems. The investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input–output system. We prove that the uniform admissibility of the pair (Cb(R,X),L1(R,X)) for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system. If p∈(1,∞) and the pair (Cb(R,X),L1(R,X)) is uniformly p-admissible then we obtain the uniform exponential trichotomy. Next, we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy. Supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input–output system has unique bounded solutions in certain subspaces. Finally we obtain that the uniform p-admissibility of the pair (Cb(R,X),L1(R,X)) is a necessary and sufficient condition for uniform exponential trichotomy.

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems

The aim of this paper is to present new discrete-time characterizations for uniform exponential trichotomy of dynamical systems. We obtain necessary and sufficient conditions for the existence of uniform exponential trichotomy of discrete dynamical systems, using a set of computational conditions of Zabczyk type. The main results are applied to deduce new criteria for the detection of the uniform exponential trichotomy of dynamical systems modeled by skew-product flows.

Homogenization for deterministic maps and multiplicative noise

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

On chain recurrence for bitransformation semigroups

Let (S, X, T) be a bitransformation semigroup. The purpose of this article is to describe the chain recurrence of (S, X) in terms of the action of T. It includes an exposition on chain transitivity of skew-product flows associated to differential systems on Lie groups.

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

We consider a general time-dependent linear competitive-cooperative tridiagonal system of differential equations in the framework of skew-product flows and obtain canonical Floquet invariant bundles which are exponentially separated. Such Floquet bundles naturally reduce to the standard Floquet space when the system is assumed to be time-periodic. We apply the Floquet theory so obtained to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

Nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces

In this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew-product flows in infinite dimensional spaces. We treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families. We show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary. Our results generalize the previous studies on this topic and provide as particular cases many interesting situations, among which we mention the detection of the exponential trichotomy of general non-autonomous systems.

Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations

Motivated by linear Schrödinger equations with almost periodic potentials and phase transitions over almost periodic lattices, we introduce the so-called skew-product quasi-flows (SPQFs), which may admit both temporal and spatial discontinuity. In this paper we establish two basic theorems for SPQFs. One is an extension of the Bogoliubov–Krylov theorem for the existence of invariant Borel probability measures and the other is the uniform ergodic theorems. As applications, it will be shown that such a Schrödinger equation admits a well-defined rotation number.

A note on diffusion limits of chaotic skew-product flows

We provide an explicit rigorous derivation of a diffusion limit—a stochastic differential equation (SDE) with additive noise—from a deterministic skew-product flow. This flow is assumed to exhibit time-scale separation and has the form of a slowly evolving system driven by a fast chaotic flow. Under mild assumptions on the fast flow, we prove convergence to a SDE as the time-scale separation grows. In contrast to existing work, we do not require the flow to have good mixing properties. As a consequence, our results incorporate a large class of fast flows, including the classical Lorenz equations.

Integral Equations and Exponential Trichotomy of Skew-Product Flows

We are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skew-product flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skew-product flows in infinite-dimensional spaces, using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations, the qualitative theory of dynamical systems in continuous time, and certain related control problems.

Translation Invariant Spaces and Asymptotic Properties of Variational Equations

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew‐product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew‐product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

Reducibility of skew-product systems with multidimensional Brjuno base flows

We develop a renormalization method that applies to the problem of the localreducibility of analytic skew-product flows on Td $\times$ SL(2,R). Weapply the method to give a proof of a reducibility theorem for these flows withBrjuno base frequency vectors.

Translation-invariant monotone systems II: Almost periodic/automorphic case

This paper studies almost periodic/automorphic monotone systems with positive translation invariance via skew-product flows. It is proved that every bounded solution of such systems is asymptotically almost periodic/automorphic. Applications are made to a chemical reaction network, especially to enzymatic futile cycles with almost periodic/automorphic reaction coefficients.

Integral equations in the study of the asymptotic behavior of skew-product flows

The aim of this paper is to present an inedit perspective concerning the study of the asymptotic behavior of variational systems, using distinct techniques compared with those in the existing literature. We give new and complete answers concerning the study of exponential dichotomy of skew-product flows in terms of the solvability of integral equations between L p -spaces, motivating the techniques by illustrative examples. We present a comparative analysis between the integral admissibility on the real line and on the half-line, pointing out the main differences as well as the advantages of each case.

Integral conditions for exponential dichotomy: A nonlinear approach

We give new necessary and sufficient integral conditions for the existence of exponential dichotomy of skew-product flows. Our methods are based on the structure of the associated stable subspace and unstable subspace. We propose a nonlinear approach, extending the study of exponential stability in terms of integral conditions to the case of uniform exponential dichotomy. We apply the main results to the study of uniform exponential dichotomy of non-autonomous systems.

Exponential trichotomy for variational difference equations

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of variational difference equations in terms of the solvability of an associated discrete-time system. First, we obtain the structure of the trichotomy projections. Then, we associate with a linear system of variational difference equations (A) an input–output system (S A ). The linear space of all sequences with finite support contained in Z + is denoted by ℱ(Z, X). We show that the uniform admissibility of the pair (ℓ∞(Z, X), ℱ(Z, X)) for the system (S A ) is a sufficient condition for the existence of uniform exponential trichotomy of the system (A). Next, we prove that the system (A) is uniformly exponentially trichotomic if and only if the pair (ℓ∞(Z, X), ℱ(Z, X)) is uniformly admissible for the associated input–output system (S A ). We also prove that the uniform exponential trichotomy of a linear skew-product flow is equivalent with the uniform exponential trichotomy of the variational difference equation associated with it. We apply our results to the study of the uniform exponential trichotomy of general linear skew-product flows in infinite-dimensional spaces.