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After introducing the notion of weighted shadowing, we give sufficient conditions under which certain nonlinear perturbations of nonautonomous linear delay differential equations exhibit the shadowing property with respect to a given weight function. As a corollary, it is shown that if the unperturbed equation admits a shifted exponential dichotomy and the Lipschitz constant of the nonlinear term is sufficiently small, then the perturbed system has a shadowing property with respect to an exponential weight function. An application to differential equations with small delay is given. The results are new even in the case of the standard shadowing property.

In this paper, block upper triangular systems of linear difference equations are considered, in which the coefficient matrices are not assumed invertible. The relationship between the exponential dichotomy properties of such a system and its associated block diagonal system is studied. The reason it is important to study triangular systems is that any system of linear difference equations is kinematically similar to an upper triangular system. In the bounded invertible case, it is known that for equations on the intervals J = Z(+) or Z(-), a block upper triangular system has an exponential dichotomy if and only if the associated block diagonal system has one. However, when J = Z, only the sufficiency holds. The sufficiency extends to the noninvertible case, provided the off-diagonal matrices are bounded. However, the necessity does not hold even when J = Z(+) or Z(-). Nevertheless, if certain conditions are added, then the necessity does hold and it is also shown that these conditions are needed since it turns out that if both the triangular and diagonal systems have dichotomies, then these extra conditions must hold.

"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."

In this work, we are interested in a class of Bi-directional associative memory neural networks (BAMNNs) with hybrid time-varying delays and D operator. We establish some sufficient conditions to guarantee the existence and global exponential stability of measure-pseudo almost periodic solutions of the considered BAMNNs, using contraction mapping principle, exponential dichotomy theory, differential inequality techniques, and the properties of the measure-pseudo almost periodic functions. A numerical example along with a graphical illustration are presented to support the effectiveness and fasibility of the obtained results. The results in this paper are original and complement the previous outcomes.

Under consideration is some class of systems of nonlinear differential equations. The exponentially dichotomous linear part of the systems is assumed to have periodic coefficients. Using the author’s criterion of exponential dichotomy, we establish the conditions of existence of periodic solutions and prove stability of the solutions under small perturbations of coefficients in the linear part and nonlinear terms.

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that ...

In this paper, we consider a class of quaternion-valued neural networks with mixed delays on time scales. By using general Lipschitz condition, contraction mapping principle and exponential dichotomy of linear dynamic equations, we prove the existence and uniqueness and the stability of almost periodic solutions of global exponential neural networks. Finally, one example is given to illustrate the efficiency of our theoretical results.

In this paper, we analyze the existence and uniqueness of remotely almost periodic solutions for systems of ordinary differential equations. The existence and uniqueness of remotely almost periodic solutions are achieved by using the results about the exponential dichotomy and the Bi-almost remotely almost periodicity of the homogeneous part of the corresponding systems of ordinary differential equations under our consideration.

This article investigates quaternion-valued fuzzy cellular neural networks with delays. With the help of fixed point theory, exponential dichotomy of linear equations and related inequalities, some new sufficient conditions which guarantee the existence and global exponentially stability of pseudo almost periodic solutions to quaternion-valued fuzzy cellular neural networks with delays are established. Computer simulations are carried out to check the practicability of the obtained main results. So far, no researchers have investigated this topic. The derived results of this article are completely innovative and complement some earlier investigations to some degree.

We study a linear cocycle over the irrational rotation $$\sigma_{\omega}(x)=x+\omega$$ of the circle $$\mathbb{T}^{1}$$ . It is supposed that the cocycle is generated by a $$C^{2}$$ -map $$A_{\varepsilon}:\mathbb{T}^{1}\to SL(2,\mathbb{R})$$ which depends on a small parameter $$\varepsilon\ll 1$$ and has the form of the Poincare map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix $$A_{\varepsilon}(x)$$ is of order $$\exp(\pm\lambda(x)/\varepsilon)$$ , where $$\lambda(x)$$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter $$\varepsilon$$ . We show that in the limit $$\varepsilon\to 0$$ the cocycle “typically” exhibits ED only if it is exponentially close to a constant cocycle. Conversely, if the cocycle is not close to a constant one, it does not possess ED, whereas the Lyapunov exponent is “typically” large.

It was proved that both the normal hyperbolicity and invariant manifold for a uniformly hyperbolic compact invariant manifold and the invariant manifold for a uniformly hyperbolic noncompact invariant manifold are persistent under small perturbation. In this paper, we weaken the uniform normal hyperbolicity to the nonuniform one and prove that both the nonuniform normal hyperbolicity and invariant manifold for a nonuniformly eventually absolutely normally hyperbolic noncompact invariant manifold are persistent under small perturbation in Banach spaces. Unlike the uniform case, we need infinitely many families of norms to uniformize infinitely many exponential dichotomies correspondingly. We use the approximative exponential dichotomy and the pseudo orbit to overcome the difficulty.

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

In this paper we give sufficient conditions on \(k\in L^1(\mathbb{R})\) and the positive measures \(\mu\), \(\nu\) such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product \(\zeta f=k\ast f\). We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of \(\left( \mu,\nu\right)\)- pseudo-almost periodic (respectively, \(\left( \mu,\nu\right)\)- pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.

En este trabajo, estudiamos la existencia de soluciones acotadas para una ecuación retardada semilineal con retardo infinito, impulso y condiciones no locales. También mostramos que bajo algunas condiciones esta solución acotada es única, periódica o casi periódica dependiendo de las condiciones impuestas a los términos que involucran la ecuación. A través de este trabajo, asumiremos que la ecuación lineal asociada tiene una dicotomía exponencial, lo que nos permite encontrar una fórmula para las soluciones acotadas, y a partir de esta fórmula, podemos aplicar el Teorema del punto fijo de Banach para demostrar la existencia de tales soluciones acotadas.

In this work, we study the existence of bounded solutions for a semilinear retarded equation with infinite delay, impulse, and non-local conditions. We also show that under some conditions this bounded solution is unique, periodic, or almost periodic depending on the conditions imposed on the terms involving the equation. Through this work, we shall assume that the associated linear equation has an exponential dichotomy, allowing us to find a formula for the bounded solutions, and from this formula, we are able to apply Banach Fixed Point Theorem to prove the existence of such bounded solutions.

We prove the existence and uniqueness of μ‐pseudo almost automorphic solution for a delayed non‐autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the φ‐Lipschitz condition and φ belongs to some admissible spaces. We further prove the existence of an invariant stable manifold around the μ‐pseudo almost automorphic solution in that case. An application is given to illustrate our theory.

We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.

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 letter 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.

We study the behavior of an exponential dichotomy for a nonautonomous linear delay equation under sufficiently small linear and nonlinear perturbations. In particular, we establish a partial version of the Grobman–Hartman theorem and we use the characterization of an exponential dichotomy in terms of an admissibility property to establish the robustness property.

This paper is concerned with an interval general Cohen-Grossberg bidirectional associative memory neural networks with mixed delays. Under proper conditions, the authors studied the existence, the uniqueness and the global exponential stability of almost automorphic solutions for the suggested system. The proposed method was mainly based on the exponential dichotomy of linear differential equation, the Banach’s fixed point principle and the differential inequality techniques. The authors illustrate with an example to demonstrate the effectiveness of the proposed findings.

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback theory, we introduce a new transformation of neutral-type equations which plays a key role in our investigation.

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

UDC 517.988.6 We obtain the necessary and sufficient conditions for the exponential dichotomy of solutions of linear difference equations with piecewise constant operator coefficients.

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation $ x^{\prime }= Ax(t) + Lx_t + f(t,x_t, \lambda ) $ , with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ.

In this article, we combine rotating periodic functions with $\mu$ ergodic functions to obtain a new class of functions called $\mu$ pseudo rotating periodic functions. Then we study the existence and uniqueness of $\mu$ pseudo rotating periodic solutions for linear systems, semi-linear systems, and non-linear systems by exponential dichotomy.

We prove the existence and uniqueness of periodic solutions to partial neutral function differential equation of the form $\frac {\partial Fu_{t}}{\partial t}= A(t)Fu_{t} +g(t,u_{t}),~t\in [0,\infty )$ , on a Banach space X, where the operator-valued function t↦A(t) and the nonlinear delay operator g(t,v) are both T-periodic with respect to t, whereas g is φ-Lipschitz with respect to v for φ belonging to an admissible space. Then, in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy, we apply our abstract results to study the existence, uniqueness of periodic solutions. We also prove the conditional stability, and the existence of local stable manifolds around such periodic solutions.

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.

The aim of this paper is to present a new and general method for the study of stability, expansiveness and dichotomy of dynamical systems, based on arguments proceeding from ergodic theory and on admissibility properties. We consider a discrete variational system and we associate to it three different families of input-output systems corresponding to the investigations for stability, expansiveness and dichotomy, respectively. We introduce some input-output conditions relative to ergodic measures and we show that certain nonuniform admissibility properties imply the existence of the uniform asymptotic behaviors. First, we obtain characterizations for uniform exponential stability. As an application, we prove that in order to preserve the uniform exponential stability subject to perturbations it is sufficient for the perturbed system to be close enough to the initial system on a measurable set. Next, we deduce criteria for uniform exponential expansiveness based on some exact admissibilities relative to ergodic measures. Finally, we present an interesting application to the study of the uniform exponential dichotomy, starting from a nonuniform admissibility property and using stability criteria.

We study a class of scalar differential equations on the circle \begin{document}$ S^1 $\end{document} . This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes \begin{document}$ \mathbb R_+ $\end{document} and \begin{document}$ \mathbb R_+ $\end{document} . Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.

The paper studies the existence of an almost periodic solution of some system of stochastic Beverton-Holt equation with higher delays and with competition between overlapping generations under some reasonable assumptions.

We consider two generalizations $$A{\cal E}_n$$ and $$W{\cal E}_n$$ of the class $${\cal E}_n$$ of linear $$n$$-dimensional exponentially dichotomous systems on the half-line. The definition of these classes differs from the classical definition of exponential dichotomy in terms of estimates for the norms of solutions in that we allow these estimates to hold starting from some time depending on the solution (the class $$A{\cal E}_n)$$ or the constant factors in the estimates to depend on the solution (the class $$W{\cal E}_n)$$. We give an equivalent definition of the class $$W{\cal E}_n$$ in the language of Bohl exponents and prove that the proper inclusions $${\cal E}_n\subset A{\cal E}_n \subset W{\cal E}_n$$ hold for $$n\geq 2$$.

For a nonautonomous linear system with nonuniform contraction, we construct a topological equivalence between this system and an unbounded nonlinear perturbation. This topological equivalence is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

The paper studies the existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equation of the form: X ( t + n ) + ∑ r = 1 n - 1 A r ( t ) X ( t + r ) + A 0 ( t ) X ( t ) = f ( t , X ( t ) ) , X\left( {t + n} \right) + \sum\limits_{r = 1}^{n - 1} {{A_r}\left( t \right)X\left( {t + r} \right) + {A_0}\left( t \right)X\left( t \right) = f\left( {t,X\left( t \right)} \right),} n ∈ 𝕑, by means of discrete dichotomy techniques.

We give new admissibility criteria for dichotomic behaviours of discrete nonautonomous systems, in infinite dimensional spaces. In the first part of the paper, we present admissibility conditions for uniform and exponential dichotomy. Next, our study is focused on polynomial dichotomy, providing new characterizations for this notion 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 $\ell^p$ -spaces and the outputs are bounded. We point out the importance of the assumptions regarding 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 application of the admissibility results we establish a robustness property of the polynomial dichotomy under small perturbations.

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equacoes diferenciais com retardamento e aplicacoes, Tese de Mestrado, Universidade de Sao Paulo, Sao Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

Let (Am)m∈Z be a sequence of bounded linear operators acting on an arbitrary Banach space X and admitting an exponential dichotomy. Further more, let fm : X → X, m ∈ Z be a sequence of Lipschitz maps. Provided that the Lipschitz constants of fm are uniformly small, we show that a nonlinear differ ence equation xm+1 = Amxm + fm(xm), m ∈ Z, exhibits various types of the weighted Hyers-Ulam stability property.

This paper considers a class of quaternion-valued Hopfield neural networks with mixed time-varying delays and leakage delays. By utilizing the exponential dichotomy of linear differential equations, Banach’s fixed point theorem and differential inequality techniques, the authors obtain some sufficient conditions to ensure the existence and global exponential stability of almost automorphic solutions for this class of quaternion-valued neural networks. The results are completely new. Finally, the authors give an example to illustrate the feasibility of the results.

The Yakubovich Frequency Theorem, in its periodic version and in its general nonautonomous extension, establishes conditions which are equivalent to the global solvability of a minimization problem of infinite horizon type, given by the integral in the positive half-line of a quadratic functional subject to a control system. It also provides the unique minimizing pair \lq\lq solution, control\rq\rq~and the value of the minimum. In this paper we establish less restrictive conditions under which the problem is partially solvable, characterize the set of initial data for which the minimum exists, and obtain its value as well a minimizing pair. The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis., Ministerio de Economía y Competitividad / FEDER, MTM2015-66330-P, Ministerio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00, European Commission, H2020-MSCA-ITN-2014, INDAM -- GNAMPA Project 2018

The article explores the questions of the initial problem and the problem of multiperiodicity solutions of linear systems integro-differential equations with an operator of the form mmctctcD∂∂++∂∂+∂∂=11τ, ()constcccm−=,,1 and with finite hereditary period 0>=conste by variable τ that describe hereditary phenomena. Along with the equation of zeros of the special differentiation operator cD are considered linear systems of homogeneous and inhomogeneous integro-differential equations, sufficient conditions are established for the unique solvability of the initial problems for them, both necessary and sufficient conditions of multiperiodic existence are obtained by ()t,τ with periods ()ωθ, of the solutions. The integral representations of multiperiodic solutions of linear inhomogeneous systems with the uniqueness property are determined 1) in the particular case when the corresponding homogeneous systems have exponential dichotomy and 2) in the general case when the homogeneous systems do not have multiperiodic solutions, except for the trivial one. The article proposes a research technique for solving problems that satisfy initial conditions and have the property of multiperiodicity with a given e heredity period for linear systems of integro-differential equations with a special partial differential operator cD. Multiperiodic solutions obtained along characteristics 00ττcctt−+= with fixed ()00,tτ are used as an application in the theory of quasiperiodic solutions of systems of integro-differential equations.

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 also show exactly when the original projection remains unchanged. Next we prove that the roughness theorem, well known for additive perturbations, holds for multiplicative perturbations also. The proof uses ideas of Zhou, Lu and Zhang. Finally, following Ducrot, Magal and Seydi, we mention that the results by Palmer on finite time conditions on dichotomy for the invertible case can be extended to the noninvertible case., The final version of the paper will appear in the Journal of difference equations and applications

Cellular neural networks with D operator and time-varying delays are found to be effective in demonstrating complex dynamic behaviors. The stability analysis of the pseudo-almost periodic solution for a novel neural network of this kind is considered in this work. A generalized class neural networks model, combining cellular neural networks and the shunting inhibitory neural networks with D operator and time-varying delays is constructed. Based on the fixed-point theory and the exponential dichotomy of linear equations, the existence and uniqueness of pseudo-almost periodic solutions are investigated. Through a suitable variable transformation, the globally exponentially stable sufficient condition of the cellular neural network is examined. Compared with previous studies on the stability of periodic solutions, the global exponential stability analysis for this work avoids constructing the complex Lyapunov functional. Therefore, the stability criteria of the pseudo-almost periodic solution for cellular neural networks in this paper are more precise and less conservative. Finally, an example is presented to illustrate the feasibility and effectiveness of our obtained theoretical results.

For a nonautonomous dynamics with discrete time, we show that if the dynamics is equivariant (respectively, reversible), then any normal form as well as the coordinate change bringing the dynamics to this normal form have equivariance (respectively, reversibility) properties. The resonances of the linear part of the dynamics are expressed in terms of the nonuniform spectrum, that in its turn is defined in terms of the notion of a tempered exponential dichotomy.

In this paper, we prove the existence and uniqueness of pseudo affine-periodic solutions for differential systems with finite or infinite delay via exponential dichotomy and some fixed point theorems. These solutions possess certain spatiotemporal structure and they might be periodic, rotating-periodic, or affine-periodic, even quasi-periodic.

In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers–Ulam stable. This class consists of equations obtained by perturbing Hill’s equation of the form $$x''=(\lambda ^2(t)-\lambda '(t))x$$ .