We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other mild assumptions, we investigate the existence, persistence and regularity of different types of invariant manifolds for these differential equations based on our previous works about invariant manifold theory for abstract `generalized dynamical systems': invariant graphs (global version) and normally hyperbolic invariant manifolds (local version); brief summaries of those works are also given.
In this paper we prove a variation of constants formula for a non autonomous and non homogeneous Cauchy problems whenever the linear part is not densely defined and is not a Hille-Yosida operator. By using this variation of constants formula we derive a necessary and sufficient conditions for the existence of exponential dichotomy for the evolution family generated by the associated non autonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally we illustrate our result by consider a parabolic equation with non local and non …
In this article we study exponential dichotomies for noninvertible linear difference equations in finite dimensions. After giving the definition, we study the extent to which the projection $P(k)$ in a dichotomy is unique. For equations on $\mathbb{Z}$ it is unique but for equations on $\mathbb{Z}_+$ only its range is unique and for $\mathbb{Z}_-$ only its nullspace. Here we strengthen Kalkbrenner's results and give a complete characterization of all possible projections. Next we study the possibility of extending the dichotomy to a larger interval. We reproduce the results of P\" otzsche but …
We introduce new weak topologies and spaces of Carathéodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carathéodory vect…
We show that non-trivial chain recurrent classes for generic $C^1$ star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability. As a result, we show that $C^1$ generic star flows have only finitely many Lyapunov stable chain recurrent classes.
We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed famil…
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by Lévy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bounded solutions. Furthermore, this unique bounded solution is almost periodic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.
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 work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential dichotomy, for which we provide several examples, study the relation with the standard notion, and establish a robustness under perturbations. We provide a dynamical interpretation of admissibility pairs related with exponential dichotomies to obtain existence of pullback and forward attractors. We apply these abstract results for ordinary and parabolic differential…
Under the condition of nonuniformly bounded growth, %nonuniform exponential dichotomy spectrum for nonautonomous linear system is proposed the relationship of the nonuniform exponential dichotomy spectrum and the other two classical spectrums (the Lyapunov spectrum and Sacker-Sell spectrum) is given, and the stability of these spectrums under small linear perturbations are summarized and presented in this paper. A main goal of this paper is to discuss the theory for the computation of these spectrums under the condition of nonuniformly bounded growth, and this extends the work of Dieci and Vle…
Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay
本文证明了具有指数二分性的紧致概自守矩阵系统的关联系统同样具有指数二分性，并揭示了其 Green 函数的紧致双概自守性，进而研究了带混合时滞的生物非自治微分方程正紧致概自守解的存在性。
In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on $\mathbb{R}^d$. Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in…
Exponential dichotomies play a central role in stability theory for dynamical systems. They allow to split the state space into two subspaces, where all trajectories in one subspace decay whereas all trajectories in the other subspace grow, uniformly and exponentially. This paper studies uniform detectability and observability notions for linear time varying systems, which admit an exponential dichotomy. The main contributions are necessary and sufficient detectability conditions for this class of systems.
In this article we introduce a nonautonomous version of the Markus--Yamabe conjecture from an exponential dichotomy spectrum point of view. We prove the validity of this conjecture for the scalar and triangular case. Additionally we show that the origin is a global attractor for an autonomous system by using nonautonomous dynamical systems tools.
We investigate singular perturbation problems caused by small delays in the view of pseudo-exponential dichotomy. For a general linear non-autonomous retarded differential equation with small delay, previous works established the existence of a pseudo-exponential dichotomy. The main objective of this paper is to give a detailed analysis of three splitting properties of this dichotomy. By obtaining serval new estimates and giving the explicit expressions of the bounds and the exponents associated with this dichotomy, we prove that as the delay tends to zero, the spectral gap approaches to infin…
In this paper we investigate some dichotomy concepts for linear difference equations in Banach spaces. We motivate our approach by illustrative examples.
We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.
In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any non-resonant condition…
In this paper, the existence conditions of nonuniform mean-square exponential dichotomy (NMS-ED) for a linear stochastic differential equation (SDE) are established. The difference of the conditions for the existence of a nonuniform dichotomy between an SDE and an ordinary differential equation (ODE) is that the first one needs an additional assumption, nonuniform Lyapunov matrix, to guarantee that the linear SDE can be transformed into a decoupled one, while the second does not. Therefore, the first main novelty of our work is that we establish some preliminary results to tackle the stochasti…
The aim of this paper is to study the normal forms of nonautonomous differential systems. For doing so, we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy, where the linear evolution operators are defined by nonautonomous differential equations $\dot x=A(t)x$ in $\R^n$. Using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations. Finally we establish the finite jet normal forms of the nonlinear differential systems $\dot x=A(t)x+f(t,x)$ in $\R^n$, which …
In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponenti…
Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.
New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))+\int_{g(t)}^t K(t,s) x(s)ds=0, $$ where $h_k(t)\leq t$, $g(t)\leq t$, $a_k(\cdot)$ and the kernel $K(\cdot,\cdot)$ are oscillatory and, generally, discontinuous functions. The proofs are based on establishing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for …
We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need …
This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.
We give a sufficient condition for existence of an exponential dichotomy for a general linear dynamical system (not necessarily invertible) in a Banach space, in discrete or continuous time. We provide applications to the backward heat equation with a potential varying in time and to the heat equation with a finite number of slowly moving potentials. We also consider the Klein-Gordon equation with a finite number of potentials whose centres move at sub-light speed with small accelerations.
We study almost automorphic solutions of the discrete delayed neutral dynamic system% \[ x(t+1)=A(t)x(t)+ΔQ(t,x(t-g(t)))+G(t,x(t),x(t-g(t))) \] by means of a fixed point theorem due to Krasnoselskii. Using discrete variant of exponential dichotomy and proving uniqueness of projector of discrete exponential dichotomy we invert the equation and obtain some limit results leading to sufficient conditions for the existence of almost automorphic solutions of the neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of inverse matrix $A\left( t\rig…
Invariant torus are constructed under assumption that the homogeneous system admits an exponential dichotomy on the semi-axes. The main result is closely related with the well-known Palmer's lemma and results of Boichuk A.A., Samoilenko A.M.
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which…
