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We investigate the roughness of nonuniform exponential dichotomies in Banach spaces subject to a new class of small linear time variable perturbations that satisfy an integral inequality which can benefit from a smallness integrability condition. We establish the continuous dependence of constants in terms of a dichotomy notion. Our proofs introduce a new development based on integral inequalities. Notably, we do not require the notion of admissibility for bounded nonlinear perturbations. Furthermore, we derive related roughness results for nonuniform exponential contractions and expansions. Our results are also new to uniform exponential dichotomy. We construct the evolution operator and projections directly, without the need for admissibility. [ABSTRACT FROM AUTHOR]

In this paper we study the unique weighted measurable shadowing property for weighted pseudo-orbits of random systems on Banach spaces with the property that linear part of random system admits a tempered exponential dichotomy. We also prove for linear random systems that the tempered exponential dichotomy is necessary for the unique weighted measurable shadowing property to hold. [ABSTRACT FROM AUTHOR]

We consider a non-autonomous coupled system (x , y) , whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C 1 invariant manifold S (t , ω) = { (x , y) ∈ X × Y | x = σ (t , y , ω) } under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation. [ABSTRACT FROM AUTHOR]

We consider the Nernst-Planck-Navier-Stokes system describing the electrodiffusion of ions in a viscous Newtonian fluid. We prove the exponential nonlinear stability of constant steady states in the case of periodic boundary conditions in any dimension of space without constraints on the number of species, valences and diffusivities. We consider also the case of two spatial dimensions, and we prove the exponential stability from arbitrary large data. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction-diffusion system with mechanical deformation of medium. This system consists of an excitable system bidirectionally coupled with an elasticity equation. The main content consists of two parts. First, for γ > 0 sufficiently small, simple pulse of homoclinic type exists. We prove that the traveling pulse is linearly stable. Specifically, there is at most one nontrivial eigenvalue near the origin and it is negative. Second, for γ > 0 large, we show existence of double twisted front-back wave loop, indicating bifurcations of various complicated traveling waves, including N -front, N -back and wave train. Then we prove that N -fronts are linearly stable. Our arguments are mainly based on geometric singular perturbation theory, exponential dichotomy, heteroclinic bifurcation and the Melnikov method. [ABSTRACT FROM AUTHOR]

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate μ in a large family of growth rates, we consider a notion of spectrum, named nonuniform μ -dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate μ , we describe all possible forms of the nonuniform μ -dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a new and complete study regarding the detection of nonuniform, uniform and exponential trichotomies of discrete dynamical systems in infinite dimensional spaces, by means of control methods of input-output type. We show for the first time that an admissibility of nonuniform nature implies the existence of a nonuniform trichotomy and we deduce the structures of the trichotomy projections. After that, we prove criteria for uniform trichotomy of discrete variational systems. Next, for p , q ∈ [ 1 , ∞ ] we introduce the notion of uniform admissibility of a (ℓ p , ℓ q) pair and we obtain complete characterizations for uniform exponential trichotomy. Throughout our study, via illustrative examples, we show that the working hypotheses are minimal and cannot be removed. All the criteria are obtained in the most general framework, in which the base space is an arbitrary set and without any additional assumption on the system coefficients. [ABSTRACT FROM AUTHOR]

To any one-sided nonautonomous dynamics obtained from a delay-difference equation, we associate a lifted autonomous delay-difference equation that can be used, together with its solutions and its perturbations, to characterize the hyperbolicity in three different ways: in terms of the hyperbolicity of the corresponding evolution map, which describes the solutions of the lifted autonomous equation; in terms of an admissibility property for this evolutions map, which corresponds to the invertibility of a certain linear operator; and in terms of an admissibility property for the perturbations of the lifted delay-difference equation. We emphasize that the evolution map is necessarily noninvertible in our setting, which led us to introduce a new notion of exponential dichotomy for this map whose existence is equivalent to the existence of an exponential dichotomy for the original nonautonomous dynamics. [ABSTRACT FROM AUTHOR]

Barreira and Valls (2010) proved in [4] the C 1 dependence of C 0 linearization on parameters for nonuniformly hyperbolic diffeomorphisms, whose nonlinear parts depend on the parameters but linear parts do not. In this paper, we further allow the linear parts to depend on the parameters, where only Hölder dependence is possible and a difficulty arises: the projections in the nonuniform exponential dichotomy do not have explicit expressions. We will overcome the difficulty by giving a new result on the Hölder dependence of nonuniform exponential dichotomy and constructing the conjugacy in a different way. Moreover, under Barreira and Valls's framework we will improve the C 1 dependence to C r , where another difficulty arises: without the non-resonant condition, the systems do not have small derivatives of orders ≥2 in general. We will introduce functional spaces with small bounds to overcome the difficulty. Besides, our different constructing method ensures that the conjugacy is invertible, which was not considered by Barreira and Valls. [ABSTRACT FROM AUTHOR]

Nonuniform exponential dichotomy represents the nonuniform hyperbolicity for evolution family. Considerable efforts have been made to the relationship between the existence of nonuniform exponential dichotomies and the admissibility of a pair of function classes. In order to reduce the effect of nonuniformity, most results on this aspect need Lyapunov norms in function classes. However, it is difficult to construct an appropriate Lyapunov norm before the existence of a nonuniform exponential dichotomy is established, which happens in the converse problem from admissibility to dichotomy. In this paper, we prove the relationship without a Lyapunov norm. Then we apply our results to giving the relationship between nonuniform exponential dichotomy and invertibility of a differential mapping on some function spaces. [ABSTRACT FROM AUTHOR]

We show that functional differential equations of mixed type (MFDEs) with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach by Hupkes and Verduyn Lunel. For the full line, we construct these splittings by generalizing the finite-range results obtained by Mallet-Paret and Verduyn Lunel. The finite dimensional space that is 'missed' by these splittings can be characterized using the Hale inner product, but the resulting degeneracy issues raise subtle questions that are much harder to resolve than in the finite-range case. Indeed, there is no direct analogue for the standard 'atomicity' condition that is typically used to rule out degeneracies, since it explicitly references the smallest and largest shifts. We construct alternative criteria that exploit finer information on the structure of the MFDE. Our results are optimal when the coefficients are cyclic with respect to appropriate shift semigroups or when the standard positivity conditions typically associated to comparison principles are satisfied. We illustrate these results with explicit examples and counter-examples that involve the Nagumo equation. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous system x ˙ = A (t) x + f (t , x , y) , y ˙ = g (t , y) and give conditions under which there is a transformation of the form H (t , x , y) = (x + h (t , x , y) , y) taking its solutions onto the solutions of the partially linearized system x ˙ = A (t) x , y ˙ = g (t , y). Shi and Xiong [28] proved a special case where g (t , y) was a linear function of y and x ˙ = A (t) x had an exponential dichotomy. Our assumptions on A and f are of the general form considered by Reinfelds and Steinberga [25] , which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove Hölder continuity of H and its inverse in x and y. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when x ˙ = A (t) x is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the L 2 -generalized solutions to the initial-boundary value problems become eventually C 2 -smooth for any initial L 2 -data. We investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on R. We prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system. Assuming that the coefficients of the hyperbolic system are almost periodic, we prove that the bounded solution is almost periodic also. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N -dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form of former's functional perturbation, and prove that the single parameter is enough to unfold all possibilities of linearly independent homoclinic solutions bifurcated from the unperturbed degenerate homoclinic one, which actually improves the known results. Furthermore, we prove that those homoclinic solutions are all transversal, showing co-existence of multiple chaotic motions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a system of non densely defined Cauchy problems and we investigate the persistence of normally hyperbolic manifolds. The notion of exponential dichotomy is used to characterize the normal hyperbolicity and a generalized Lyapunov-Perron approach is used in order to prove our main result. The result presented in this article extend the previous results on the center manifold by allowing a nonlinear dynamic in the unperturbed central part of the system. We consider two examples to illustrate our results. The first example is a parabolic equation coupled with an ODE that can be considered as an interaction between an antimicrobial and bacteria while the second one is a Ross-Macdonald epidemic model with age of infection. In both examples we were able to reduce the infinite dimensional system to an ordinary differential equation. [ABSTRACT FROM AUTHOR]

Abstract We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil. [ABSTRACT FROM AUTHOR]

We investigate evolution families generated by general linear first-order hyperbolic systems in one space dimension with periodic boundary conditions. We state explicit conditions on the coefficient functions that are sufficient for the existence of exponential dichotomies on R in the space of continuous periodic functions. [ABSTRACT FROM AUTHOR]

Relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems. It was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy. In this paper we find an appropriate admissible pair of classes of Lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half-lines R ± separately, on both half-lines R ± simultaneously, and on the whole line R . Additionally, the maximal admissibility is proved in the case on both half-lines R ± simultaneously. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems. The investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input–output system. We prove that the uniform admissibility of the pair ( C b ( R , X ) , L 1 ( R , X ) ) for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system. If p ∈ ( 1 , ∞ ) and the pair ( C b ( R , X ) , L 1 ( R , X ) ) is uniformly p -admissible then we obtain the uniform exponential trichotomy. Next, we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy. Supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input–output system has unique bounded solutions in certain subspaces. Finally we obtain that the uniform p -admissibility of the pair ( C b ( R , X ) , L 1 ( R , X ) ) is a necessary and sufficient condition for uniform exponential trichotomy. [ABSTRACT FROM AUTHOR]

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is asymptotically stable. As in [14,24,25] , we show that the sequence of traveling waves is itself asymptotically stable: as t → ∞ , the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Our proof is similar to that of [14] in that it is based on spatial dynamics, Laplace transform, and exponential dichotomies, but it incorporates a number of modifications. [ABSTRACT FROM AUTHOR]

Abstract: For nonautonomous dynamical systems a bifurcation can be understood as topological change in the set of bounded entire solutions to a given time-dependent evolutionary equation. Following this idea, a Fredholm theory via exponential dichotomies on semiaxes enables us to employ tools from analytical branching theory yielding nonautonomous versions of fold, transcritical and pitchfork patterns. This approach imposes the serious hypothesis that precise quantitative information on the dichotomies is required — an assumption hard to satisfy in applications. Thus, imperfect bifurcations become important. In this paper, we discuss persistence and changes in the previously mentioned bifurcation scenarios by including an additional perturbation parameter. While the unperturbed case captures the above bifurcation patterns, we obtain their unfolding and therefore the local branching picture in a whole neighborhood of the system. Using an operator formulation of parabolic differential, Carathéodory differential and difference equations, this will be achieved on the basis of recent abstract analytical techniques due to Shi (1999) and Liu, Shi and Wang (2007). [Copyright &y& Elsevier]

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: 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]

Abstract: Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor–repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space. [Copyright &y& Elsevier]

Abstract: Consider an evolution family on a half-line and a semi-linear integral equation . We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging 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. The existence of such manifolds is obtained 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 certain classes of admissible function spaces. [Copyright &y& Elsevier]

Abstract: Hyperbolicity of an autonomous rest point is characterised by its linearization not having eigenvalues on the imaginary axis. More generally, hyperbolicity of any solution which exists for all times can be defined by means of Lyapunov exponents or exponential dichotomies. We go one step further and introduce a meaningful notion of hyperbolicity for linear systems which are defined for finite time only, i.e. on a compact time interval. Hyperbolicity now describes the transient dynamics on that interval. In this framework, we provide a definition of finite-time spectrum, study its relations with classical concepts, and prove an analogue of the Sacker–Sell spectral theorem: For a d-dimensional system the spectrum is non-empty and consists of at most d disjoint (and often compact) intervals. An example illustrates that the corresponding spectral manifolds may not be unique, which in turn leads to several challenging questions. [Copyright &y& Elsevier]

Abstract: We prove that the operator G, the closure of the first-order differential operator on , is Fredholm if and only if the not well-posed equation , , has exponential dichotomies on and and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, , A is the generator of a bi-semigroup, is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems. [Copyright &y& Elsevier]

Abstract: In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system and the nonlinear system . Some sufficient conditions are obtained to guarantee the existence of a equivalent function sending the -quasibounded solutions of nonlinear system onto those of linear system . Our results generalize the Palmer''s linearization theorem in [K.J. Palmer, A generalization of Hartman''s linearization theorem, J. Math. Anal. Appl. 41 (1973) 753–758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191]. Moreover, we prove that equivalent function is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191] never considered this important property of the equivalent function . [Copyright &y& Elsevier]

Abstract: We study the Fredholm properties of parabolic evolution equations on with inhomogeneous boundary values. These problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces. Assuming that the underlying homogeneous problem is asymptotically hyperbolic, we show the Fredholm alternative for these equations. The results are applied to parabolic partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors. [Copyright &y& Elsevier]

Abstract: In this paper we consider the singular value decomposition (SVD) of a fundamental matrix solution in order to approximate the Lyapunov and exponential dichotomy spectra of a given system. One of our main results is to prove that SVD techniques are sound approaches for systems with stable and distinct Lyapunov exponents. We also show how the information which emerges with the SVD techniques can be used to obtain information on the growth directions associated to given spectral intervals. [Copyright &y& Elsevier]

Abstract: A very general characterization of exponential dichotomy for evolutionary processes in terms of the admissibility of some pair of spaces which are translation invariant (the so-called Schäffer spaces) is given. It includes, as particular cases, many interesting situations among which we note the results obtained by N. van Minh, F. Räbiger and R. Schnaubelt and the authors concerning the connections between admissibility and dichotomy. [Copyright &y& Elsevier]

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on R with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both R+ and R- and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators A(t),t∈R, the operator G is a closure of the operator - d/dt+A(t). Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg. [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]

A functional differential equation admitting a complete guiding set always has an a priori bounded solution. The introduction of this concept and the idea of reducibility allow the authors to apply a well-known method for the autonomous case to a non-autonomous problem. A previous result of Ortega and Tineo is extended to a non-hyperbolic semilinear functional differential equation with almost-periodic coefficients in its lineal part. The non-hyperbolicity condition is established in terms of the Sacker–Sell spectrum of the associated linear system. [Copyright &y& Elsevier]

In the present work, we give necessary and sufficient conditions for linear differential non-autonomous equations on a Banach space, that satisfy exponential dichotomy, to be topologically equivalent. We shall also prove that such equations are structurally stable. [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]

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]