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We obtain a Datko-type characterization of several classes of a strong nonuniform exponential behaviour. We consider separately the case of exponential contractions, expansions and dichotomies. In addition, we consider both discrete and continuous time. As a nontrivial application of our work, we study the robustness property of strong nonuniform exponential contractions, expansions and dichotomies. [ABSTRACT FROM AUTHOR]

In this paper, a numerical algorithm for computing the connecting orbits in piecewise smooth dynamical systems is derived and is analyzed. A nondegenerate condition for the connecting orbit with respect to its bifurcation parameter is presented to ensure the defining equation being well posed, which is a generalization of the Melnikov condition for smooth systems. The error caused by the truncation of time interval is also analyzed. Some numerical calculations are carried out to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show that in the study of dichotomy for evolutionary family generated by differential systems, using the Perron's method, it is enough to consider space L∞ as an output, because when the output space is Lq, q < ∞, it can be used the Datko's theorem (see P. Preda, M. Megan [17]). [ABSTRACT FROM AUTHOR]

In this paper, we study a relation between exponential dichotomy on time scales and admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for an evolution family on time scales. We establish a sufficient criterion for the existence of exponential dichotomy on time scales in terms of the admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for the evolution family. Conversely, with the help of exponential dichotomy on time scales, we give the admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for an input-output equation on time scales. [ABSTRACT FROM AUTHOR]

We study an asymptotic behavior of solutions of a nonlinear system by comparing with a linear system, whose asymptotic behavior of solutions is similar to the behavior of solutions of the original system. [ABSTRACT FROM AUTHOR]

This paper deals with the inversion problem for linear time-invariant dynamical systems. Earlier in this formulation the discussed issue has been addressed and solved for invertible systems that meet the definition of Isidori relative degree. But in fact, this problem is well-posed for a wider range of linear dynamical systems. In this paper we examine a particular case based on a definition that is similar to the definition of relative degree by Isidori. Presented here columnwise relative degree and its properties allow to consider and effectively solve the inversion problem for invertible systems that meet this definition. [ABSTRACT FROM AUTHOR]

This paper gives a version of Hartman-Grobman theorem for the impulsive differential equations. We assume that the linear impulsive system has a nonuniform exponential dichotomy. Under some suitable conditions, we proved that the nonlinear impulsive system is topologically conjugated to its linear system. Indeed, we do construct the topologically equivalent function (the transformation). Moreover, the method to prove the topological conjugacy is quite different from those in previous works (e.g., see Barreira and Valls, 2006). [ABSTRACT FROM AUTHOR]

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t,x)| ≤ μe-ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t,x)| ≤ μe-ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be |f(t,x)| ≤ μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption |f(t,x)| ≤ μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper. [ABSTRACT FROM AUTHOR]

We prove that a family of q-periodic continuous matrix valued function {A(t)}t∈R has an exponential dichotomy with a projector P if and only if ∫0t eiμsU(t, s)Pds is bounded uniformly with respect to the parameter μ and the solution of the Cauchy operator Problem ...(t) = -Y (t)A(t) + eiμt(I - P), t ≥ s Y (s) = 0, has a limit in L(Cn) as s tends to ∞1 which is bounded uniformly with respect to the parameter μ. Here, {U(t, s) : t, ∈ R} is the evolution family generated by {A(t)}t∈R, μ is a real number and q is a fixed positive number. INSET: X. [ABSTRACT FROM AUTHOR]

Carathéodory differential equations naturally occur as path-wise realization of random differential equations and are amenable for deterministic calculus. In the setup of such nonautonomous differential equations with only measurable time-dependence, we present an approach to a bifurcation theory based on a topological change in the set of bounded entire solutions. In such a setting of at least planar equations, we provide sufficient criteria for a nonhyperbolic entire solution to bifurcate into two branches of bounded or homoclinic solutions. As opposed to transcritical or pitchfork bifurcations, no trivial solution branch is supposed to exist in advance. In particular, we discuss a degenerate fold bifurcation pattern, where the transversality assumption is replaced by a nondegeneracy condition on the second-order derivative. Both bifurcation patterns are intrinsically nonautonomous and do not occur for time-invariant equations. Our notion of a nonhyperbolic solution is based on the fact that the associate variational equation possesses exponential dichotomies on both semiaxes with compatible projectors. The resulting Fredholm theory allows one to apply recent abstract bifurcation results due to Liu, Shi and Wang (2007). [ABSTRACT FROM AUTHOR]

Abstract: We obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations assuming that satisfy “Acquistapace–Terreni” conditions, the evolution family generated by has exponential dichotomy, is almost periodic, are densely defined closed linear operators, are Lipschitz with respect to the second argument uniformly in the first argument, is pseudo almost periodic in the first argument, is Stepanov-like pseudo almost periodic in the first argument for and jointly continuous. To illustrate our abstract results, two examples are given. [Copyright &y& Elsevier]

By means of the fixed point methods and the properties of the pseudo almost automorphic functions, the existence and uniqueness of pseudo almost automorphic solutions are obtained for differential equations involving reflection of the argument. For the nonscalar, case we use the exponential dichotomy properties. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study weighted pseudo almost periodic solutions of neutral functional differential equations. By applying the properties of weighted pseudo almost periodic functions and the exponential dichotomy of linear systems as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of weighted pseudo-almost periodic solution of the equations. [Copyright &y& Elsevier]

We obtain necessary and sufficient conditions for the hyperbolicity of a semigroup of operators. In so doing, we use Lyapunov's equation in operator form constructed from its generator. [ABSTRACT FROM AUTHOR]

Abstract: We analyze the existence of (eventually no past) exponential and ordinary dichotomies of an exponentially bounded, strongly continuous cocycle over a continuous semiflow σ. Our main tool is the -admissibility condition (i.e. there exist , , such that for each input and , there exists such that the output belongs to ). We prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to θ. Requiring that the boundedness to be uniform with respect to θ, we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for . Variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained. It is worth to note that we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the well-known concept defined by Sacker and Sell (1994) . Our definition of exponential dichotomy follows partially the definition given in Chow and Leiva (1996) in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to Perron (1930) , Daleckij and Krein (1974) , Massera and Schäffer (1966) , Nguyen van Minh, Räbiger and R. Schnaubelt (1998) . [Copyright &y& Elsevier]

Abstract: We study a nonlinear neutral functional differential equation. Applying the properties of almost periodic function and exponential dichotomy of linear system as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of almost periodic solution of the equations. [Copyright &y& Elsevier]

Abstract: We give new necessary and sufficient integral conditions for the existence of exponential dichotomy of skew-product flows. Our methods are based on the structure of the associated stable subspace and unstable subspace. We propose a nonlinear approach, extending the study of exponential stability in terms of integral conditions to the case of uniform exponential dichotomy. We apply the main results to the study of uniform exponential dichotomy of non-autonomous systems. [Copyright &y& Elsevier]

Abstract: Using the method of discretization, we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew-product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear-skew product semiflows. We then apply the obtained results to consider the roughness of exponential dichotomy of linear-skew product semiflows. [Copyright &y& Elsevier]

Abstract: In this work we show when and how techniques based on the singular value decomposition (SVD) and the QR decomposition of a fundamental matrix solution can be used to infer if a system enjoys—or not—exponential dichotomy on the whole real line. [Copyright &y& Elsevier]

In this paper, we discuss the existence of almost periodic solutions for a class of nonlinear Duffing equations. Some new theorems are established to ensure that the existence of almost periodic solutions for the above Duffing equations. Our results generalize some recently results (see Remark 2.3). [ABSTRACT FROM AUTHOR]

Abstract: We study exponential dichotomy of linear skew-product semiflows which come from linearizing skew-product semiflows on a compact positively invariant subset of semiflows and construct the relationship between continuous separation and exponential dichotomy under assumptions that skew-product semiflows are eventually strongly monotone. In addition, we deduce that the exponential dichotomy is trivial when is hyperbolically stable, and the hyperbolic instability of is the necessary condition of the state space admitting a trivial separation in another forms. Simultaneously, we list some conditions for hyperbolic stability and instability of . At last, we construct a sufficient and necessary condition for exponential dichotomy of linear skew-product semiflows in terms of the admissibility of the pair . [Copyright &y& Elsevier]

Abstract: We show that if is an exponentially dichotomic evolution operator, then the unique solution of the Volterra equation is also an exponentially dichotomic evolution operator, for small . As a consequence, we prove that any exponentially dichotomic evolution family is structurally stable. We improve the previous results, showing that the condition of bounded growth and decay, present everywhere in the existing literature, can be completely removed. [Copyright &y& Elsevier]

Abstract: In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations and in Banach spaces. As we will see, our results allow for a more general to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to even if the evolution family generated by is exponentially stable and . [Copyright &y& Elsevier]

Abstract: Consider an evolution family on a half-line and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that has an exponential dichotomy and the nonlinear forcing term satisfies the non-uniform Lipschitz conditions: for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of type, the Lorentz spaces and many other function spaces occurring in interpolation theory. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of -almost-periodic solutions for partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition. We use the reduction principle developed recently in [M. Adimy, K. Ezzinbi, A. Ouhinou, Variation of constants formula and almost-periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications 317 (2006) 668–689] to prove the existence of a -almost- periodic solution when there is at least one bounded solution in . We give an application to the Lotka–Volterra model with diffusion. [Copyright &y& Elsevier]

Abstract: This paper is concerned with pseudo-almost periodicity of the solutions to the nonautonomous evolution equation with delay . Some sufficient conditions which ensure the existence and uniqueness of pseudo-almost periodic mild solutions to the evolution equation with delay are given. An example is shown to illustrate our results. [Copyright &y& Elsevier]

Abstract: Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals. [Copyright &y& Elsevier]

Equations of retarded type and simple neutral-type equations are considered. The study concerns both autonomous and non-autonomous perturbations of an autonomous equation which possesses a non-trivial periodic orbit. The main tool is a local coordinate system around the periodic orbit which is obtained from the phase space decomposition via Floquet multipliers. Under the assumption that the perturbation function is Lipschitz the existence of an integral manifold with periodic structure for the system in the new coordinates is shown. This implies that, under autonomous perturbations, periodic orbits are continued. Furthermore, we give a description of the flow on the center manifold of the periodic orbit. [Copyright &y& Elsevier]