Polynomial Dichotomy Research Articles

On uniform dichotomy in mean of stochastic skew-evolution semiflows in Banach spaces

Abstract In this paper we consider three concepts of uniform dichotomy in mean for stochastic skew-evolution semiflows: uniform exponential dichotomy in mean, uniform polynomial dichotomy in mean and uniform h- dichotomy in mean. Some characterizations of these notions and connections between these concepts are given. The obtained results can be considered generalizations to the dichotomy case of the results from [29].

Nonuniform Dichotomy with Growth Rates of Skew-Evolution Cocycles in Banach Spaces

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented.

Barbashin type characterizations for nonuniform <i>h</i>-dichotomy of evolution families

<abstract> <p>The aim of this paper is to give some Barbashin type characterizations for the nonuniform <italic>h</italic>-dichotomy of reversible evolution families in Banach spaces. Two necessary and sufficient conditions for the nonuniform <italic>h</italic>-dichotomy are pointed out using some important sets of growth functions. Additionally, as particular cases, we obtain a Barbashin type characterization for nonuniform exponential dichotomy and a necessary and sufficient condition for the nonuniform polynomial dichotomy.</p> </abstract>

On Some Characterizations for Uniform Dichotomy of Evolution Operators in Banach Spaces

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established.

"Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems"

"We give new admissibility criteria for dichotomic behaviours of discrete nonautonomous sys- tems, in infinite dimensional spaces. First, we present admissibility conditions for uniform and exponential dichotomy. Next, our study is focused on polynomial dichotomy, providing new characterizations for this no- tion by means of some double admissibilities. We obtain two categories of criteria for polynomial dichotomy, based on input-output conditions imposed to some suitable systems such that, for each one, the input sequences belong to certain $l^p$ -spaces and the outputs are bounded. We point out the importance of the assumptions re- garding the complementarity of the stable subspaces at the initial time and we also discuss the relevance of the concept of solvability (unique or not) in the admissibility criteria for polynomial dichotomies on the half-line. All the results are obtained in the general case, without any additional hypotheses on the systems coefficients and without assuming any growth type properties for the associated propagators. Furthermore, as an applica- tion of the admissibility results we establish a robustness property of the polynomial dichotomy under small perturbations"

"On Polynomial Dichotomies of Discrete Nonautonomous Systems on the Half-Line"

"The aim of this paper is to provide new characterizations for polynomial dichotomies of discrete nonautonomous systems on the half-line. First, we establish equivalent structures for the ranges of projections for a polynomial dichotomy with respect to a sequence of norms. Next, we establish the connections between polynomial dichotomies and other dichotomic behaviors. We obtain for the first time a characterization of polynomial dichotomy with respect to a sequence of norms in terms of ordinary dichotomy and exponential dichotomy of suitable systems with respect to well-chosen sequences of norms. The results are obtained in the most general case, without any additional assumptions regarding the coefficients of the underlying systems."

Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

Datko type characterizations for nonuniform polynomial dichotomy

The aim of the present paper is to give two characterization theorems of Datko type for the nonuniform polynomial dichotomy concept with respect to invariant projection families and also with respect to strongly invariant projection families.

Admissibility and generalized nonuniform dichotomies for discrete dynamics

<p style='text-indent:20px;'>We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.</p>

Admissibility and general dichotomies for evolution families

For an arbitrary noninvertible evolution family on the half-line and for ρ : [0, ∞) → [0, ∞) in a large class of rate functions, we consider the notion of a ρdichotomy with respect to a family of norms and characterize it in terms of two admissibility conditions. In particular, our results are applicable to exponential as well as polynomial dichotomies with respect to a family of norms. As a nontrivial application of our work, we establish the robustness of general nonuniform dichotomies.

NONUNIFORM POLYNOMIAL DICHOTOMY WITH LYAPUNOV TYPE NORMS

The paper considers a general concept of polynomial dichotomy which includes as particular cases some well-known dichotomy concepts. The main objective is to obtain some characterizations of the nonuniform polynomial dichotomy behavior with respect to a family of norms compatible with the projection families

Admissibility and nonuniform polynomial dichotomies

AbstractFor a general one‐sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. As a nontrivial application, we establish the robustness of the notion of a nonuniform polynomial dichotomy.

Admissibility and polynomial dichotomies for evolution families

For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.

Uniform Polynomial Dichotomy for Linear Discrete-time Systems

The aim of this paper is to give some characterizations of the uniform polynomial dichotomy for linear discrete dynamical systems on the half-line.

Integral conditions for nonuniform $μ$-dichotomy on the half-line

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

The (k,ℓ) unpartitioned probe problem NP-complete versus polynomial dichotomy

A graph G=(V,E) is Cprobe if V can be partitioned into two sets, probes P and non-probes N, where N is independent and new edges may be added between non-probes such that the resulting graph is in the graph class C. We say that (N,P) is a Cprobe partition for G. The Cunpartitioned probe problem consists of an input graph G and the question: Is G a C probe graph? A (k,ℓ)-partition of a graph G is a partition of its vertex set into at most k independent sets and ℓ cliques. A graph is (k,ℓ) if it has a (k,ℓ)-partition. We prove the full complexity dichotomy into NP-complete and polynomial for (k,ℓ)unpartitioned probe problems: they are NP-complete if k+ℓ≥3, and polynomial otherwise. This gives the first examples of graph classes C that can be recognized in polynomial time but whose probe graph classes are NP-complete.

Nonuniform Polynomial Dichotomy for Noninvertible Linear Discrete-Time Systems in Banach Spaces

We study three polynomial dichotomy concepts for linear discrete-time systems in Banach spaces. Our main objective is to give characterizations in terms of Lyapunov functions for nonuniform polynomial dichotomy of nonautonomous and noninvertible linear discrete-time systems.

Generalized Nonuniform Dichotomies and Local Stable Manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.

Lyapunov Functions for General Nonuniform Dichotomies

For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.

Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x t + f (t, x t , λ), assuming that the linear equation x′ = L(t)x t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.