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Low-dimensional and Non-autonomous Dynamics

Final Report Summary - LDNAD (Low-dimensional and Non-autonomous Dynamics)

This research project aimed at making significant contributions to the bifurcation theory for non-autonomous (i.e. forced or random) dynamical systems. Particular focus was placed on studying several open problems and questions in low dimensions.

Bifurcation theory is the study of the qualitative changes that can occur in the dynamics of a system when varying parameters. Bifurcations can occur in both discrete systems (described by maps) and continuous time systems (generated by differential equations). Several local bifurcations have been studied in the classical theory, including, the saddle node bifurcation, transcritical, pitchfork, period doubling, and Hopf bifurcation.

An important question which arises frequently in applications is that of the influence of external forcing on the bifurcation patterns of dynamical systems. However, despite this relevance to applications and significant progress in recent years, our understanding of non-autonomous bifurcations is still incomplete, even in the low-dimensional context.

This research project aimed at making contributions to the bifurcation theory for non-autonomous dynamical systems, and in particular, to the non-autonomous counterparts of the classical bifurcation patterns in low dimensions. The research objectives included the development of a non-autonomous bifurcation theory for deterministic dynamical systems, the development of a general qualitative theory for forced monotone interval maps with transitive forcing, the development of a bifurcation theory for random dynamical systems, the description and rigorous analysis of the Hopf bifurcation. During the project, we developed insights and tools in order to complement the study of low-dimensional non-autonomous bifurcation theory.

At the start of the project, some of our work on the non-autonomous Hopf bifurcation was completed. In particular, we studied a general class of model systems which exhibit the full two-step scenario for the non-autonomous bifurcation, a long standing open problem in the field (proposed by Arnold). The scenario was described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. Here, external forcing can lead to a separation of the complex conjugate eigenvalues, giving rise to the two-step bifurcation scenario, in which an invariant 'torus' splits off a previously stable central manifold. In particular, we proved that the split-off 'torus' consists of a topological circle in each fibre. Up to now, this description was mainly based in numerical evidence and no non-trivial examples existed for which this bifurcation pattern was described analytically. We gave a description of the non-autonomous bifurcation in a class of model systems which is accessible to rigorous analysis, but at the same time allows for highly non-trivial dynamics.
We also derived analogues for continuous-time models generated by planar vector fields.

Subsequently, we studied two dimensional systems of ordinary differential equations where a complex conjugate pair of eigenvalues of the linearised flow at the equilibrium become purely imaginary, exhibiting a Hopf bifurcation for the system. Several different types of non-autonomous counterparts of this system were analysed, and for each of these systems, the highly non-trivial pullback attractor was studied numerically and described analytically.

In addition, significant progress was made in optimization questions in ergodic theory. In particular, we considered the full shift; the lexicographic order induces a partial order (known as first-order stochastic dominance) on the collection of its shift-invariant probability measures. We studied the fine structure of this dominance order, gave conditions for comparability, and proved that the Sturmian measures (supported on the periodic and aperiodic sequences of minimal complexity) are totally ordered with respect to this order.