## Workshop on Stability of Dynamical Systems

There will be a workshop at the University of Exeter on Tuesday 27th March 2012. There is no registration fee for this workshop but we would appreciate knowing if you will attend so we can organize catering. We thank the Royal Society for funding the visit of Dr Podvigina. Please let or know if you want to attend or need more information.

A provisional schedule is:

**Until 4pm we will be in the room Washington Singer 219**

10.30-11.00 Martin Krupa (Paris/Exeter) **Mixed-mode oscillations in a multiple time scale phantom bursting.**

*Abstract :* We present some results on Mixed Mode Oscillations (MMOs) in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow null-cline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow null-cline through a fold point of the fast null-cline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.

11.00-11.30 Jonathan Dawes (Bath)** Periodically-perturbed robust heteroclinic cycles.
**

*Abstract*: We investigate the effect of time-periodic perturbation terms on the dynamics near the traditional Guckenheimer-Holmes robust heteroclinic cycle. There are two natural ways of incorporating periodicity: ‘externally’, as additive perturbations terms or ‘internally’ through periodically-variation in the eigenvalues at the saddle-type equilibria. The dynamics can be quantitatively analysed through the construction of return maps, and explained in terms of the dynamics of circle maps andplanar ODEs in various asymptotic limits. This is joint work with Tsung-Lung Long Tsai (NCUE, Taiwan).

11.30-12.00 Alexander Lohse (Hamburg) **On relations between the stability index and attraction properties of heteroclinic cycles**

In my talk I will present recent results on the relationship between the stability index for heteroclinic cycles that Peter Ashwin and Olga Podvigina defined in their paper “On local attraction properties and a stability index for heteroclinic connections” and different stability properties of the cycle. The index quanties the extent of the basin of attraction of the cycle in a small ε-ball around it. It is related to stability properties of the cycle, but not always in an obvious way. For example, if there is a point on the cycle where the index exists and is greater than 0, then the cycle is essentially asymptotically stable (e.a.s.). A similar result can be shown for predominant asymptotic stability (p.a.s.) of the cycle in the case that the index exists everywhere and is greater than some constant c > 0. On the way to proving these results we establish a more general statement, namely that any cycle with a set of positive measure in its basin of attraction is e.a.s. One crucial idea for the proof of this was developed by Olga Podvigina.

12.10-13.30 lunch

13.30-14.10 Mike Field (Houston) **Dynamics, adaptivity and asynchronous logic in large networks of coupled dynamical systems.**

We discuss a class of large adaptive networks of coupled discrete dynamical systems that relate to problems in computational neuroscience and learning (notably, Spike-Timing Dependent Plasticity or STDP). We describe some interesting examples of heteroclinic phenomena and cycling chaos that occur in these networks as well as some issues related to synchronous and asynchronous logic. We also show some preliminary work on SLOGALS logic (SLOppy Globally Asynchronous Locally Synchronous logic) that has implications for large asynchronous networks with significant random components.

14.10-14.40 Chris Bick (Goettingen) Dynamics and Bifurcations in Symmetric Phase Oscillator Systems”

14.40-15.30 Tea

**Move to Poldhu Room, Kay Building**

16.05-16.55 Olga Podvigina (Moscow) **Stability and bifurcations of heteroclinic cycles of type Z (ACCESS GRID DYNAMICS SEMINAR)
**

Dynamical systems which are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this talk I discuss stability properties of a class of structurally stable heteroclinic cycles called heteroclinic cycles of type Z. It is well-known that a heteroclinic cycle that is not asymptotically stable can attract nevertheless a positive measure set from its small neighbourhood. I call such cycles fragmentarily asymptotically stable. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearisations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, I discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.