Attractors

Limitations of pullback attractors for processes

(with P.E. Kloeden and M. Rasmussen) Journal of Difference Equations and Applications 18(4), 2012, 693-701

Pullback convergence has been investigated in numerous papers as an appropriate attraction concept for nonautonomous problems. However, in this note we indicate through some simple examples of nonautonomous difference equations that pullback attractors do not give a complete picture of asymptotic behaviour when the nonautonomous dynamical systems that they generate are formulated as processes. It is then shown how the problems can be resolved by using a skew-product formulation of the nonautonomous dynamical systems.

Dedicated to Francisco Balibrea on the occasion of his 60th birthday.

Morse decompositions for delay-difference
equations

(with Á. Garab), Journal of Dynamics and Differential Equations 31(2) (2019), 903-932

Scalar difference equations $x_{k+1}=f(x_k,x_{k-d})$ with delay $d\in\N$ are well-motivated from applications e.g. in the life sciences or discretizations of delay-differential equations. We investigate their global dynamics by providing a (nontrivial) Morse decomposition of the global attractor. Under an appropriate feedback condition on the second variable of $f$, our basic tool is an integer-valued Lyapunov functional.

Forward and pullback dynamics of nonautonomous integrodifference equations: Basic constructions

(with Huy Huynh and P.E. Kloeden), Journal of Dynamics and Differential Equations 34 (2020), 671-699

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback and forward attractors, as well as forward limit sets are constructed fro general infinite-dimensional nonautonomous dynamical systems in discrete time. While the theory of pullback attractors, but not their application to integrodifference equations, is meanwhile well-established, the present ovel approach is needed in order to understand their future behavior.