Journal
Since 1 February 2013 I am Lecturer (Assistant Professor) at the Department of Mathematical Sciences of Xi’an Jiaotong - Liverpool University.
TLDR: https://efstathiou.gr/unlinked/standard-map.
In the context of an introductory lecture on Hamiltonian systems it is interesting to demonstrate several phenomena related to perturbing an integrable Hamiltonian system: Poincaré-Birkhoff chains, stability islands, the persistence and destruction of invariant tori, the consequent transition to global stochasticity, homoclinic and heteroclinic tangles and the Smale horseshoe, and many more. The standard map is a great playground for exploring these phenomena.
Nevertheless, presenting only a static sequence of phase portraits would be a great injustice to the richness of the dynamical behavior that can be exhibited by Hamiltonian dynamical systems in general, and the standard map in particular. It is much better to let people explore such systems by themselves through an interactive program.
The workshop Spectral Analysis, Stability and Bifurcation in Modern Nonlinear Physical Systems took place at the Banff International Research Station between 4 and 9 November 2012. In the workshop I gave an invited talk on the recent work about using covering maps to prove fractional monodromy in 2-DOF integrable Hamiltonian systems in \(n_1{:}-n_2\) resonance. The video of the presentation is available from the BIRS website.
Our paper with Andrea Giacobbe on cusp singularities of integrable Hamiltonian fibrations has now been accepted for publication in Nonlinearity. Andrea and I tried to understand how cusp singularities fit together in the momentum domain of integrable Hamiltonian fibrations. It turns out that when two cusp singularities are connected by a curve of hyperbolic singularities there are locally two possible topologies of the unfolded momentum domain that we call pleat and flap. The pleat topology is associated to bidromy while the flap topology is associated to non-trivial Hamiltonian monodromy. For more details you can check the paper.
The paper on efficient structure-aware selection techniques for 3D point clouds that will be presented at Visweek'12, taking place in Seattle, was selected for a Honorable Mention award.
George Efstathiou was born on 15 August 2012 at 13:36 at the Martini Hospital in Groningen. His weight was 2.63kg and his length 48cm. Lingyun and George are both doing fine.
Occasionally I do things that fall outside my main research interests. One of these things is a recent work with Lingyun Yu, Petra Isenberg, and Tobias Isenberg where we tried to develop intuitive computer algorithms for the interactive selection of particle clusters in 3D particle datasets. We were motivated in this effort by the problem of selecting clusters in N-body galaxy simulations but the algorithms we developed can be applied in other domains that deal with 3D particle datasets.
The conference 5th Shanghai International Symposium on Nonlinear Sciences and Applications took place last week at the Fudan University in Shanghai and on a cruise from Yichang to Chongqing along the Yangtze river.
In the conference I gave an invited talk with title «Synchronization in biological systems» where I presented the work [1,2] on pulse-coupled oscillator systems with delay I have done in collaboration with Easwar Subramanian and Henk Broer.
In the period 7-17 May 2012 I visited the Université de Bourgogne in Dijon for a collaboration with Robert Roussarie on the regularization of discontinuous dynamical systems.
On December 19-20, 2011 the Brain-Mind workshop took place at Fudan University in Shanghai. In the workshop I presented one of the invited keynotes. The title of my talk was “Applications of dynamical systems in biology and synchronization”.
The concept of synchronization plays a very important role in biology. In the talk I presented 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 non-zero delay the state space of such systems is infinite dimensional. Important questions here are the existence of unstable attractors, i.e., of saddle periodic orbits whose stable set has non-empty interior. In an earlier work we showed that for any number \(n\) of oscillators with \(n \ge 3\) there is an open parameter region in which the system has unstable attractors. Moreover, in the case of \(n = 4\) oscillators we showed that there exist unstable attractors with heteroclinic cycles between them. The second such system is a model for circadian rhythms. We have studied how a single pacer cell synchronizes to a periodic signal. This signal includes the effect of the external environment (light-dark 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. In the presentation I discussed the properties of this family (emphasizing resonances and Arnol’d tongues) and their biological significance.
Recently Domien Beersma, Henk Broer, Kim Gargar, Igor Hoveijn, and I published in Physica D a paper on synchronization and circadian rhythms. The basic idea behind the paper is to study in a simple model how a single pacer cell synchronizes to a periodic signal. This signal includes the effect of the external environment (light-dark 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. In the paper we discuss the properties of this family (emphasizing resonances and Arnol’d tongues) and their biological significance.
The conference Symmetry and Perturbation Theory 2011 took place last week in Otranto. The conference was organized by Giuseppe Gaeta, Ferdinand Verhulst, Raffaele Vitolo, and Sebastian Walcher.
In the conference I gave an invited talk with title «Uncovering fractional monodromy» presenting a work I am doing in collaboration with Henk Broer. The main idea behind this work is that when dealing with systems that have fractional monodromy it is more natural to first lift them to a branched covering space. Such lift simplifies the geometry of the fibers and allows a straightforward description of the global geometry. Pushing this description down to the original space reveals the fractional monodromy.