Research
The chair works on nonautonomous dynamical systems, that is, on systems whose rules change over time. For autonomous systems the theory is well developed. As soon as time enters, many of the classical notions lose their foundation. That is where the work begins. Which notions of spectrum, hyperbolicity, normal form and stability remain viable when the system itself changes, and what takes the place of those that do not?
Research lines
Spectral theory and dichotomies of nonautonomous systems
For time-dependent equations it is not obvious what a spectrum should even be. This work develops the dichotomy spectrum as an answer and probes how far it carries: in nonuniform form, for difference equations, on time scales. The most recent line is the Bohl dichotomy, a notion that applies where the exponential dichotomy fails. Joint with Adam Czornik (Gliwice) and Konrad Kitzing.
Control theory and stabilisation of time-varying systems
How much of the spectrum can feedback prescribe in a time-varying system, and when does local information suffice? The question of assignability of the dichotomy spectrum links spectral theory to control engineering. It is at the heart of the collaboration with the Silesian University of Technology in Gliwice.
Bifurcation theory and nonautonomous normal forms
Normal forms, linearisation and smoothing when the passage of time is itself part of the problem. The classical theorems of Hartman-Grobman, Poincaré and Sternberg are carried over to the nonautonomous case. Invariant fibre bundles take the place of invariant manifolds.
Finite-time dynamics and Lyapunov methods
Asymptotic theory asks what happens as t → ∞. What one can observe, however, is always a finite window of time. This line develops notions of hyperbolicity, entropy and spectrum that do without the limit. It thereby finds coherent structures in fluid flows that the classical theory does not see.
Fractional differential equations and Hilbert space methods
Equations with memory, cast functional-analytically: stability, invariant manifolds and well-posedness of fractional systems in Hilbert space. Joint work with Rainer Picard (TU Dresden), Sascha Trostorff (Kiel), Marcus Waurick (Freiberg), Kai Diethelm, and the group around Doan Thai Son at the Institute of Mathematics of the Vietnam Academy of Science and Technology in Hanoi.
Applications: biology, networks, electrodynamics
The basic research reaches into concrete models: pest control and dengue transmission, networks of coupled systems and their stability, a storm protection system for buildings. Added to this is a geometric re-reading of Maxwell's equations.
Collaborations
Standing partners of the chair are:
- Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi. Doan Thai Son (Director of the Institute, PhD at TU Dresden in 2009), Nguyen Dinh Cong, Hoang The Tuan and Luu Hoang Duc (currently at the Max Planck Institute for Mathematics in the Sciences in Leipzig): fractional differential equations, finite-time dynamics, random dynamical systems
- Silesian University of Technology, Gliwice. Adam Czornik, Artur Babiarz and Michał Niezabitowski (Institute of Automatic Control): spectral and control theory of time-varying systems
- Escuela Superior Politécnica del Litoral, Guayaquil. Joseph Páez Chávez: numerical bifurcation analysis and applications in biology and engineering
- University of West Bohemia, Pilsen. Petr Stehlík: dynamical systems on graphs and time scales
- TU Bergakademie Freiberg and Kiel University. Marcus Waurick and Sascha Trostorff: Hilbert space methods for evolutionary equations
Publications
The work on these research lines is arranged by the same lines on the publications page. The complete list is continuously updated from the research information system of TU Dresden.