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### Open Access Journals

DCDS

Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surprising because such a behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques.

Based on operator-theoretical tools, this paper provides various sufficient and verifiable criteria to obtain a corresponding

Based on operator-theoretical tools, this paper provides various sufficient and verifiable criteria to obtain a corresponding

*diagonal significance*for finite-dimensional difference equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.
DCDS

This paper continues our work on local bifurcations for nonautonomous difference and ordinary differential equations. Here, it is our premise that constant or periodic solutions are replaced by bounded entire solutions as bifurcating objects in order to encounter right-hand sides with an arbitrary time dependence.

We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.

We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.

DCDS-B

We investigate local bifurcation properties for nonautonomous difference and ordinary differential equations. Extending a well-established autonomous theory, due to our arbitrary time dependence, equilibria or periodic solutions typically do not exist and are replaced by bounded complete solutions as possible bifurcating objects.

Under this premise, appropriate exponential dichotomies in the variational equation along a nonhyperbolic solution on both time axes provide the necessary Fredholm theory in order to employ a Lyapunov-Schmidt reduction. Among other results, this yields nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns.

Under this premise, appropriate exponential dichotomies in the variational equation along a nonhyperbolic solution on both time axes provide the necessary Fredholm theory in order to employ a Lyapunov-Schmidt reduction. Among other results, this yields nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns.

DCDS-B

As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.

DCDS-S

Topological linearization results typically require solution flows rather than merely semiflows. An exception occurs when the linearization fulfills spectral assumptions met e.g. for scalar reaction-diffusion equations. We employ tools from the geometric theory of nonautonomous dynamical systems in order to extend earlier work by Lu [12] to time-variant evolution equations under corresponding conditions on the Sacker-Sell spectrum of the linear part. Our abstract results are applied to nonautonomous reaction-diffusion and convection equations.

CPAA

We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem.

As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.

As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.

DCDS

We prove a necessary and sufficient condition for the exponential
stability of time-invariant linear systems on time scales in terms of
the eigenvalues of the system matrix. In particular, this unifies the
corresponding characterizations for finite-dimensional differential
and difference equations. To this end we use a representation formula
for the transition matrix of Jordan reducible systems in the
regressive case. Also we give conditions under which the obtained
characterizations can be exactly calculated and explicitly calculate
the region of stability for several examples.

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