Wykład: Nonautonomous Dynamics: Attractors, Manifolds, Stability, Spectra

Minicourse On Nonautonomous Stability

2017-05-23, godz. 10:15, Aula Wydziału Matematyki i Informatyki

The theory of dynamical systems has seen a remarkable progress over the last 100 years, beginning with the contributions of Poincaré and Lyapunov to a contemporary detailed understanding of the attractor for various infinite-dimensional systems. This success is partly due to the restriction to autonomous systems. However, many real-world problems are actually nonautonomous. That is, they involve time-dependent parameters, controls, modulation and various other effects. Special cases include periodically or almost periodically forced systems, but in principle the time dependence can be arbitrary. As a consequence, many of the now well-established concepts, methods and results for autonomous systems are no longer applicable and require appropriate extensions.

In this (of course biased) mini-course, we discuss several basic ingredients from the theory of nonautonomous difference equations. Among them are pullback convergence, exponential dichotomies and the related dichotomy spectrum, which form the basis for a stability theory and nonautonomous invariant manifolds.

The prerequisite for the course is a little knowledge (and interest) in the classical theory of (autonomous) dynamical systems.

References:
[1] A.N. Carvalho, J.A. Lang and J.C. Robinson: Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer, Heidelberg etc., 2012
[2] P.E. Kloeden and C. Pötzsche: Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics (Mathematical Biosciences Subseries) 2102, Springer, Heidelberg etc., 2013
[3] P.E. Kloeden and M. Rasmussen: Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, AMS, Providence, RI, 2011.
[4] P. Kloeden, C. Pötzsche and M. Rasmussen. Discrete-time nonautonomous dynamical systems. In R. Johnson and M. Pera, editors, Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lect. Notes Math. 2065, 35–102. Springer, Heidelberg, 2012.
[5] C. Pötzsche: Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, in E. Liz, editor, Proceedings of the workshop on future directions in difference equations, Vigo, Spain, June 13–17, 2011, Coleccion Congresoson Congresos 69, 163–212, Vigo, 2011. Servizo de Publicacións de Universidade de Vigo.

Wykładowca: prof. Christian Pötzsche, Alpen-Adria Universität Klagenfurt, Austria