Bernd Aulbach

The dichotomy spectrum for noninvertible systems of linear difference equations

In this paper we introduce the so—called dichotomy spectrum for nonautonomous linear difference equations , whose coefficient matrices are not supposed to be invertible. This new kind of spectrum is based on the notion of exponential forward dichotomy and consists of at most N+ 1 closed, not necessarily bounded, intervals of the positive real line. If all the matrices A(k) are invertible then the number of spectral intervals is at most N, and if in addition A(k) ≡ A is independent of k then these intervals reduce to the absolut values of the eigenvalues of A. In any case the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate. The main result of this paper is a Spectral Theorem which describes all possible forms of the dichotomy spectrumIn this paper we introduce the so—called dichotomy spectrum for nonautonomous linear difference equations , whose coefficient matrices are not supposed to be invertible. This new kind of spectrum i...

The concept of spectral dichotomy for linear difference equations II

This paper is a continuation of a previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept of spectral dichotomy has been introduced. This new notion of dichotomy has proved to be useful since it allows to apply the well known theory of linear operators to study dynamic properties of nonautonomous linear difference equations. In the present paper we extend our result on the equivalence of the spectral dichotomy and the well known exponential dichotomy to the class of linear differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate equations on the set of positive integers for which we establish necessary and sufficient conditions for exponential and unifrom stability.

The fundamental existence theorem on invariant fiber bundles

For discrete dynamical systems the theory of invariant manifolds is well known to be of vital importance. In terms of difference equations this theory is basically concerned with autonomous equations. However, the crucial and currently most difficult questions in this field are related to non-periodic, in particular chaotic motions. Since this topic - even in the autonomous context is an intrinsically time-variant matter. There is and urgent need for a non-autonomous version of invariant manifold theory. In this paper we present we present a very general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms. In fact, we drop the assumption of invertibility of the mapping, we consider non-autonomous difference equations rather than mappings In effect, we generalize the notion of invariant manifold to the concept of invariant fiber bundle.

Nonlinear semigroups and the existence and stability of solutionsof semilinear nonautonomous evolution equations

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

Asymptotic stability regions via extensions of Zubov's method—I

On considere un systeme differentiel x˙=f(t,x) satisfaisant les hypotheses suivantes:f:R n+1 →R n est analytique; f(t,x) est presque periodique en t pour chaque x∈R n , f(t,0)=0 ∀t∈R; la matrice principale fondamentale Ψ(t,s) de y˙=f x (t,0)y satisfait ∥Ψ(t,s)∥≤βexp−γ(t−s) ∀t≥s avec β et γ des constantes positives; f(t,x)−f x (t,o)x=0(∥x∥ uniformement en t quand ∥x∥→0

A Smoothness Theorem for Invariant Fiber Bundles

Invariant fiber bundles are the generalization of invariant manifolds from discrete dynamical systems (mappings) to non-autonomous difference equations. In this paper we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called “Hadamard–Perron-Theorem” for time-dependent families of pseudo-hyperbolic mappings from the finite-dimensional invertible to the infinite-dimensional non-invertible case.

An Elementary Proof for Hyperbolicity and Chaos of the Logistic Maps

It is well known, that for any the maximal compact invariant set of the logistic map is hyperbolic and that is chaotic on Λμ. While this result can be found in most books dealing with chaotic dynamics, its proof is usually given only for In this paper, we present an elementary and self-contained proof for all

Linearizing the expanding part of noninvertible mappings

The main result of this paper is on partial linearization by means of a topological transformation of a mapping which is not supposed to be invertible. Our approach also provides a new proof (based on elementary degree theory) of the Hartman-Grobman Lemma as well as sharp results on the smoothness of the pseudo-stable foliation. The results are valid in arbitrary Banach spaces.

Invariant manifolds with asymptotic phase

THIS PAPER is concerned with invariant manifolds allowing a generic approach to solutions (of the underlying autonomous differential system) only. This means. that any solution approaching such an invariant manifold M ultimately behaves like a particular solution on J4, in the sense that the difference n-(t) i(l) of the two solutions tends to 0 as t + CQ. In other words, only solutions on the stable manifold of M can approach M as f+ cc. Invariant manifolds with that property are said to have an asymptotic phase. The simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose Floquet multipliers lie all but one in the open unit disc of the complex plane. Malkin [12] proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold. More generally, a k-dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k Floquet multipliers have modulus less than 1 (see Hale & Stokes [9]). Conditions for a torus to be asymptotically stable with asymptotic phase are given by Samoilenko [13] and Coppel[5]. Invariant manifolds without any specified structure are considered by Fenichel [6,7] and in Kirchgraber’s note [ 111. Moreover, each center manifold has an asymptotic phase if the corresponding equilibrium is stable (see Carr & Al-Amood [4] and Henry [lo]). In any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold. In each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability. A negative answer to this question is given by the author in [l, 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase. In [l] the manifold is supposed to be compact while the noncompact case is treated in [2] under the restriction that the generating family of periodic solutions has an amplitude independent period. Recently Hale & Massatt [8] extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations. In this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result: A compact invariant manifold has an asymptotic phase if the manifold is hyperbolic (assumption Hl) and carries a flow parallel in the sense of our assumption H2. This includes most of the aforementioned results.

Partial linearization for noninvertible mappings

A Hartman-Grobman result for noninvertible mappings is proved. It is assumed that the spectrum of the linearized mapping contains zero but is disjoint from the complex unit circle. In the infinite-dimensional case, additional spectral conditions are assumed. These additional conditions are satisfied if the linearized mapping is completely continuous. The proof combines HartmansC1 linearization method for contractive mappings [10] with our previous result [1] on linearization of the expanding part.