Christian Pötzsche

Chain rule and invariance principle on measure chains

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

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.

Discrete-Time Nonautonomous Dynamical Systems

These notes present and discuss various aspects of the recent theory for time-dependent difference equations giving rise to nonautonomous dynamical systems on general metric spaces:First, basic concepts of autonomous difference equations and discrete-time (semi-) dynamical systems are reviewed for later contrast in the nonautonomous case. Then time-dependent difference equations or discrete-time nonautonomous dynamical systems are formulated as processes and as skew products. Their attractors including invariants sets, entire solutions, as well as the concepts of pullback attraction and pullback absorbing sets are introduced for both formulations. In particular, the limitations of pullback attractors for processes is highlighted. Beyond that Lyapunov functions for pullback attractors are discussed.Two bifurcation concepts for nonautonomous difference equations will be introduced, namely attractor and solution bifurcations.Finally, random difference equations and discrete-time random dynamical systems are investigated using random attractors and invariant measures.

Reducibility of linear dynamic equations on measure chains

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.

Integral manifolds under explicit variable time-step discretization

We study the behavior of the “full hierarchy” of integral manifolds, i.e. in particular those of stable, center-stable, center, center-unstable and unstable type, for nonautonomous ordinary differential equations in Banach spaces under explicit one-step discretization with varying step-sizes. Our main results on C m − 1-closeness under such discretizations are formulated in a quantitative fashion and turn out to be an easy consequence of a general theorem on the existence of invariant fiber bundles within the “calculus on time scales”.

Invariant foliations and stability in critical cases

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.

Pseudo-stable and Pseudo-unstable Fiber Bundles for Dynamic Equations on Measure Chains

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.