Skew-product Flows Research Articles

Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li–Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl ( 2 , R ) -valued, continuous or discrete cocycle. Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.

On Exponential Dichotomy of Variational Difference Equations

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew‐product flows.

On Dichotomous Behavior of Variational Difference Equations and Applications

We give new and very general characterizations for uniform exponential dichotomy of variational difference equations in terms of the admissibility of pairs of sequence spaces overℕwith respect to an associated control system. We establish in the variational case the connections between the admissibility of certain pairs of sequence spaces overℕand the admissibility of the corresponding pairs of sequence spaces overℤ. We apply our results to the study of the existence of exponential dichotomy of linear skew-product flows.

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ℝ d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.

Carrying simplices in nonautonomous and random competitive Kolmogorov systems

The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system.

Dynamical Order in Systems of Coupled Noisy Oscillators

We investigate a dynamical order induced by coupling and/or noise in systems of coupled oscillators. The dynamical order is referred to a one-dimensional topological structure of the global attractor of the system in the context of random skew-product flows. We show that if the coupling is sufficiently strong, then the system exhibits one dimensional dynamics regardless of the strength of noise. If the coupling is weak, then it is shown numerically that the system also exhibits one dimensional dynamics provided the noise is sufficiently strong. We also show that for any coupling and any noise, the system has a unique rotation number and hence all the oscillators tend to oscillate with the same frequency eventually (frequency locking).

Lyapunov's second method for nonautonomous differential equations

Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems.

Resummation of Perturbation Series and Reducibility for Bryuno Skew-Product Flows

We consider skew-product systems on 핥d ×l SL(2,ℝ) for Bryuno base flows close to constant coefficients, depending on a parameter, in any dimension d, and we prove reducibility for a large measure set of values of the parameter. The proof is based on a resummation procedure of the formal power series for the conjugation, and uses techniques of renormalisation group in quantum field theory.

Lyapunov and Dynamical Spectra for Banach State-spaces

We study the Lyapunov exponents and exponential splittings for continuous linear skew-product flows acting on infinite dimensional Banach state-spaces.

A normal form theorem for Brjuno skew systems through renormalization

We develop a dynamical renormalization method for the problem of local reducibility of analytic linear quasi-periodic skew-product flows on T 2 × SL ( 2 , R ) . This approach is based on the continued fraction expansion of the linear base flow and deals with “small-divisors” by turning them into “large-divisors.” We use this method to give a new proof of a normal form theorem for a Brjuno base flow.

Flattening, squeezing and the existence of random attractors

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

New criteria for exponential expansiveness of variational difference equations

The aim of this paper is to obtain general input–output conditions for uniform exponential expansiveness of variational difference equations in terms of the complete admissibility of pairs of sequence spaces. We introduce a large class Q ( N ) of Banach sequence spaces and we deduce the connections between the complete admissibility of the pair ( B ( Θ , V ( N , X ) ) , U ( N , X ) ) with U , V ∈ Q ( N ) and the uniform exponential expansiveness of a system of variational difference equations. We apply our results at the study of the uniform exponential expansiveness of linear skew-product flows.

New criteria for exponential stability of variational difference equations

The aim of this work is to obtain very general characterizations for uniform exponential stability of variational difference equations, using Banach sequence spaces. We prove that a system of variational difference equations is uniformly exponentially stable if and only if there is a Banach sequence space B , with certain properties, such that the set of all vectors with the corresponding orbits contained uniformly in B is of the second category. We apply our result at the study of the uniform exponential stability of linear skew-product flows.

Input-output conditions for the asymptotic behavior of linear skew-product flows and applications

In this paper we present a new approach concerning the uniform exponential dichotomy of linear skew-product flows and extend existing results on exponential dichotomy roughness for variational systems in infinite dimensional spaces. We introduce new concepts of admissibility and we deduce their connections with the uniform exponential dichotomy of discrete linear skew-product flows. We apply our results at the study of the exponential dichotomy roughness of discrete linear skew-product flows, presenting an estimation for the lower bound of the dichotomy radius.

ON THE CONTINUITY OF LIAO QUALITATIVE FUNCTIONS OF DIFFERENTIAL SYSTEMS AND APPLICATIONS

Let 𝔛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let [Formula: see text], 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 𝔛r(M), Liao defined functions [Formula: see text], k = 1, …, ℓ, on [Formula: see text], which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every [Formula: see text] depends Cr-1-continuously upon [Formula: see text] and Cr-continuously on [Formula: see text] for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system.

A Generalization of a Theorem of Liao

Let X be a metrizable space and let ϕ:ℝ × X → X be a continuous flow on X. For any given {φt}–invariant Borel probability measure, this paper presents a {ϕt}–invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the continuous flow (X, {ϕt}). The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non–uniformly hyperbolic systems and skew–product flows.

Exponential dichotomy and expansivity

In this work we show if a linear nonautonomous system of ordinary differential equations satisfies a bounded growth condition, then it is exponentially expansive (a slightly stronger condition than uniformly noncritical as defined by Massera and Schaffer) on a half-line if and only if it has an exponential dichotomy and on a whole line if and only if it has an exponential dichotomy on both half-lines and no nontrivial bounded solution. We relate these theorems to Sacker and Sell’s work on linear skew-product flows and also examine the robustness of exponential expansivity under perturbation.

Global and pullback attractors of set-valued skew product flows

We investigate the asymptotic behaviour of a general set-valued skew product flow (SVSPF), that is, a set-valued cocycle mapping (coming from a non-autonomous differential equation or inclusion) driven by another, autonomous, system. Absorptivity conditions which ensure the existence of several types of attractors for such set-valued systems are established. The topological properties of and relations between these attractors, in forward and pullback senses and their strong and weak versions are analyzed. Several illustrative examples are also provided.

Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair \((C_0 ({\mathbf{R}},X),C_0 ({\mathbf{R}},X)).\) As a consequence, we prove that a linear skew-product flow π on \(E = X \times \Theta \) is uniformly exponentially dichotomic if and only if the pair \((C_0 ({\mathbf{R}},X),C_0 ({\mathbf{R}},X))\) is uniformly admissible for π.

Exponential stability for linear skew-product flows

We give general characterizations for uniform exponential stability of linear skew-product flows. We present a unified treatment for discrete and integral conditions for uniform exponential stability. As applications, for the particular case of evolution families, we generalize some results due to Przyluski, Rolewicz and Zabczyk.