Oscillator networks, including neuronal ensembles, can exhibit multiple cooperative rhythms such as chimera and cluster states. However, understanding which rhythm prevails remains challenging. In this talk, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. We show that three-cluster splay states with two distinct, coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units.

The unipotent flow on the unit tangent bundle of the modular surface is a classic example of a homogeneous flow when understood through the identification with P SL2(R)/P SL2(Z). The ergodicity of the flow implies that almost every orbit is dense in the space and hence must eventually make excursions deeper and deeper into the cusp. We are interested in understanding the nature of these excursions.

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild
assumptions we prove the existence of a unique invariant mixing absolutely continuous
probability measure, study its rate of decay of correlation and prove a number of limit
theorems.

What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)?  This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem.  I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics and transfer operato

Dynamical systems generated by scalar reaction-diffusion equations enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system.

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact.