Dynamics on time scales

Extended hierarchies of invariant fiber bundles

Differential Equations and Dynamical Systems 18(1-2), 2010,
105-133

If a linear autonomous ordinary differential of difference equation possesses a coefficient operator, which is (pseudo-) hyperbolic or allows a more specific splitting of its spectrum into appropriate spectral sets, then this gives rise to a so-called hierarchy of invariant linear subspaces of X related to the ranges to the corresponding spectral projections. Together with the intersections of these invariant subspaces, we get an extended hierarchy. Here, each member of the hierarchy can be characterized dynamically as set of initial points for orbits with a certain asymptotic growth rate in forward or backward time.

In this paper we show that such a scenario persists under perturbations w.r.t. two points of view: In the first instance, the invariant linear spaces become an “extended hierarchy” of invariant manifolds, if the linear part is perturbed by a globally Lipschitzian (or smooth) mapping on X. This will be done in the nonautonomous context of dynamic equations on measure chains or time scales, where the time-varying invariant manifolds are called invariant fiber bundles. Secondly, we derive perturbation results well-suited for up-coming applications in analytical discretization theory.

This paper is dedicated to the memory of Prof. Bernd Aulbach, without whom this hardly would have been possible.

Topological decoupling, linearization and perturbation on inhomogeneous time scales

Journal of Differential Equations 245, 2008, 1210-1242

We derive a linearization theorem in the framework of dynamic equations on time scales. This extends a recent result from [Y. Xia, J. Cao & M. Han, A new analytical method for the linearization of dynamic equation on measure chains, Journal of Differential Equations 235 (2007), 527–543] in various directions: Firstly, in our setting the linear part needs not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global Hartman-Grobman theorem for nonautonomous problems. Secondly, we investigate the behavior of the topological conjugacy under parameter variation.

These perturbation results are tailor-made for future applications in analytical discretization theory, i.e., to study the relationship between ODEs and numerical schemes applied to them.

Invariant foliations and stability in critical cases

Advances in Difference Equations 2006, 1-19

We construct invariant foliations of the extended state space for nonautonomous semilinear dynamic equations on measure chains (time scales). These equations allow a specific parameter dependence which is the key to obtain perturbation results necessary for applications to an analytical discretization theory of ODEs. Using these invariant foliations we deduce a version of the Pliss reduction principle.

Delay equations on measure chains:

Basics and linearized stability

in Proceedings of the 8th International Conference on Difference Equations and Applications, B. Aulbach, O. Dosly, S. Elaydi, G. Ladas, eds., Chapman & Hall/CRC, Boca Raton 2005, 227-234

We introduce the notion of a delay dynamic equation, which includes differential and difference equations with possibly time-dependent backward delays. After proving a basic global existence and uniqueness theorem for appropriate initial value problems, we derive a criterion for the asymptotic stability of such equations in case of bounded delays.

Cm-smoothness of invariant fiber bundles for dynamic equations on measure chains

(with S. Siegmund) Advances in Difference Equations 2, 2004, 141-182

We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem“ to the time-dependent, infinite-dimensional, noninvertible, and parameter-dependent case, where the linear part is not necessarily hyperbolic with variable growth rates. As a key feature, our proof works without using complicated technical tools.

On periodic dynamic equations on measure chains

Dynamic Systems and Applications 4, 2004, 422-428

We prove some basic results on the existence of periodic solutions for linear and non-linear dynamic equations on measure chains and time scales.

Two perturbation results for semi-linear dynamic equations on measure chains

in Proceedings of the 6th International Conference on Difference Equations and Applications, B. Aulbach, S. Elaydi, G. Ladas, eds., Chapman & Hall/CRC, Boca Raton 2004, 325-335

In this note we investigate semi-linear parameter dependent dynamic equations on Banach spaces and provide sufficient criteria for them to posses exponentially bounded solutions in forward and backward time. Apart from classical stability theory these results can be applied in the construction of non-autonomous invariant manifolds.

Pseudo-stable and pseudo-unstable fiber bundles for dynamic equations on measure chains

Journal of Difference Equations and Applications 9(10), 2003, 947-968

Invariant fiber bundles are the generalization of invariant manifolds from classical discrete or continuous dynamical systems to non-autonomous dynamic equations on measure chains. 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 hyperbolic finite-dimensional diffeomorphisms to pseudo-hyperbolic time-dependent non-regressive dynamic equations in Banach spaces. The proof of their smoothness uses a fixed point theorem of Vanderbauwhede-Van Gils.

A spectral characterization of exponential stability for linear time-invariant systems on time scales

(with S. Siegmund and F. Wirth) Discrete and Continuous Dynamical Systems 9(5), 2003, 1223-1241

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.

Reducibility of linear dynamic equations on measure chains

(with B. Aulbach), Journal of Computational and Applied Mathematics 141, 2002, 101-115

The concepts of reducibility and kinematic similarity are of major significance in the theory of stability of linear differential and difference equations. In this paper, we generalize some fundamental results on reducibility from the finite-dimensional differential equations context to dynamic equations on measure chains in arbitrary Hilbert spaces. In fact, we derive sufficient conditions for dynamic equations to be kinematically similar to an equation with zero right-hand side or to an equation in Hermitian or block diagonal form.

Chain rule and invariance principle on measure chains

Journal of Computational and Applied Mathematics 141, 2002, 249-254

In this note we prove a chain rule for mappings on abstract measure chains and apply our result to deduce an invariance principle for nonautonomous dynamic equations.

Exponential dichotomies for dynamic equations on measure chains

Nonlinear Analysis (TMA) 47(2), 2001, 873-884

In this paper we introduce the notion of an exponential dichotomy for not necessarily invertible linear dynamic equations in Banach spaces within the framework of the “Calculus on measure chains“. In particular, this unifies the corresponding theories for difference and differential equations. We apply our approach to obtain results on perturbed systems.