Linear Skew-product Flows Research Articles

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.

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.

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.

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.

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.

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.

Exponential stability and exponential instability for linear skew-product flows

We give characterizations for uniform exponential stability and uniform exponential instability of linear skew-product flows in terms of Banach sequence spaces and Banach function spaces, respectively. We present a unified approach for uniform exponential stability and uniform exponential instability of linear skew-product flows, extending some stability theorems due to Neerven, Datko, Zabczyk and Rolewicz.

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c 0(N, X), c 0(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c 0(N, X), c 0(N, X)) and (C 0(R +, X), C 0(R +, X)), respectively. We generalize an expansiveness theorem due to Van Minh, Rabiger and Schnaubelt, for the case of linear skew-product flows.

The Principal Spectrum for Linear Nonautonomous Parabolic PDEs of Second Order: Basic Properties

We put forward a theory extending the notion of principal eigenfunction and principal eigenvalue to the case of linear nonautonomous parabolic PDEs of second orderut=∑i, j=1Naij(t, x)∂2u∂xi∂xj+∑i=1Nai(t, x)∂u∂xi+a0(t, x)u, x∈Ω,on a bounded domain Ω⊂RN, with Dirichlet or Robin boundary conditions. A canonically defined one-dimensional subbundle S (corresponding to the solutions that are globally defined and of the same sign) serves as an analog of principal eigenfunction. The principal spectrum is defined to be the dynamical (Sacker–Sell) spectrum of the linear skew-product flow on S. Characterizations of principal spectrum in terms of (logarithmic) growth rates of positive solutions are given. Finally, monotonicity of the principal spectrum with respect to zero order terms is proved.

A New Approach to Asymptotic Diagonalization of Linear Differential Systems

We study the asymptotic diagonalization of a system consisting of an $$L_{{\text{loc}}}^p $$ -matrix plus a finite number of $$L^{m_i } $$ -perturbations on an interval I 0=[t 0, ∞), where p, m i∈[1, ∞). Using linear skew-product flows and spectral theory, we show that if the unperturbed system has full spectrum over its omega-limit set, then the entire system is asymptotically diagonalizable almost everywhere.

On Growth Rates of Subadditive Functions for Semiflows

Letφ:X×T+→Xbe a semiflow on a compact metric spaceX. A functionF:X×T+→Xis subadditive with respect toφifF(x, t+s)⩽F(x, t)+F(φ(x, t),nbsp;s). We define the maximal growth rate ofFto be supx∈Xlimsupt→∞(1/t)F(x, t). This growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow. Applications to Birkhoff sums, characteristic exponents of linear skew-product semiflows on Banach bundles, and average Lyapunov functions are developed. In particular, a relationship between the dynamical spectrum and the measurable spectrum of a linear skew-product flow established by R. A. Johnson, K. J. Palmer, and G. R. Sell (SIAM J. Math. Anal.18, 1987, 1–33) is extended to semiflows in an infinite dimensional setting.

Hyperbolic Linear Skew-Product Semiflows

A spectral theory for evolution operators on Banach spaces has been developed in [14, 15] considering associated C_0 -semigroups on vector-valued function spaces. It is then quite natural to substitute the shift on \mathbb R by an arbitrary flow \sigma on a topological space X and to substitute the evolution operator by a cocycle \Phi over \sigma . This task was performed by Latushkin and Stepin (cf. [8, 9]) for hyperbolic linear skew-product flows assuming some norm continuity of this flow. In general only strong continuity can be obtained (cf. Sacker and Sell [18) and Example 2 below). Following a suggestion by Hale [7: p. 601 we consider strongly continuous linear skew-product flows in Banach spaces and characterize hyperbolicity through a spectral condition.

Spectral properties of weighted composition operators and hyperbolicity of linear skew-product flows

Spectral properties of weighted composition operators and hyperbolicity of linear skew-product flows