LDNAD · Low-dimensional and Non-autonomous Dynamics

This research project aims at making significant contributions to the bifurcation theory for non-autonomous (i.e., forced or random) dynamical systems.Dynamical systems is a very active research field, with a plethora of applications in other areas of mathematics as well as the applied sciences. Many dynamical systems arising from real-world applications are forced (non-autonomous), that is, driven by some external system or noise. In recent decades, there has been steadily growing interest in the theory of non-autonomous dynamical systems, which was mainly motivated by applications in physics, biology, engineering, chemistry, economics, ecology and other disciplines.Mathematical modelling is used extensively in engineering, and the natural and social sciences and typically gives rise to complicated dynamical systems depending on one or several parameters. Fluctuations in these physical parameters can lead to qualitative changes in the behaviour of the system (when a parameter reaches a critical value), referred to as a bifurcation or critical transition, where a sudden change in the dynamics is observed.Bifurcations and critical transitions occur in a wide variety of applications including climate change, medicine, and economics, and the understanding of the dynamical behaviour of systems near bifurcation points plays an important role to control and attenuate the expected consequences.The main aim of this research project, is to develop insights and tools in order to complement the study of non-autonomous bifurcation theory. The proposal contains the following research directions:1. The development of a non-autonomous bifurcation theory for deterministic dynamical systems.2. The development of a general qualitative theory for forced monotone interval maps with transitive forcing.3. The development of a bifurcation theory for random dynamical systems.4. The description and rigorous analysis of the stochastic Hopf bifurcation.