February 10-14, 2014
Self Similar Solutions to Curve Shortening in Rn
Keywords of the presentation: Curve Shortening, Self Similar solutions, Morse Decomposition
Curves that evolve under Curve Shortening by a combination of rescaling and Euclidean motions are solutions to a system of ordinary differential equations on the unit tangent bundle of Rn.
The flow, which has no fixed points, does admit an interesting Morse decomposition, especially after compactifying the phase space.
In this talk I will present the many different solutions to Curve Shortening that arise in this way.
Semigroups and the Study of Coupled Cell Networks
Dynamical systems with a coupled cell network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web.
A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier.
This is joint work with Jan Sanders.
Braided Solutions of Differential Equations
Keywords of the presentation: braids, Morse-Conley-Floer homology, computer assistence
Pieces of string or curves in three dimensional space may be knotted or
braided. This topological tool can be used to study certain types of nonlinear
differential equations. In particular, such an approach leads to forcing
theorems in the spirit of the famous "period three implies chaos" for interval
maps. We have identified three types of differential equations that respect the braid structure, and for these we can construct Morse-Conley-Floer type (homological) invariants. I will also illustrate how the topological arguments can be combined with a computer-assisted approach.