Title: Applications of dynamical systems in biology and synchronization
Speaker: Konstantinos Efstathiou
Researcher, Department of Mathematics, University of Groningen, Netherlands
Time: 14:10 am—14:50am, December 20 (Tuesday)
Venue: Room 102, SubBuilding of East Guanghua Tower
Abstract:
The concept of synchronization plays a very important role in biology. I will present two systems that exhibit synchronization. The first such system is a network of pulse coupled oscillators with delay. Such networks are used for modelling, for example, the activity in biological neuron networks or the synchronization processes in networks of interacting agents. Because of the nonzero delay the state space of such systems is infinite dimensional. We study the existence of unstable attractors, i.e., of saddle periodic orbits whose stable set has nonempty interior. We prove that for any number of oscillators n >= 3 there is an open parameter region in which the system has unstable attractors. Moreover, in the case of n=4 oscillators we show that there exist unstable attractors with heteroclinic cycles between them. The second such system is a model for circadian rhythms. We study how a single pacer cell synchronizes to an periodic signal. This signal includes the effect of the external environment (lightdark cycle) but also the effect of the rest of the pacer cells. It turns out that such system can be described by a family of circle maps. We discuss the properties of this family (emphasizing resonances and Arnol’d tongues) and their biological significance.
