Journal

Filippov systems are dynamical systems with discontinuities. They are being used to model the dynamics in cases where there is very rapid transition between different dynamical regimes. For example, if in a predator-prey system the number of prey crosses a threshold then the predation rate can increase discontinuously.

As the simplest case of a Filippov system one can consider a two-dimensional dynamical system where the phase space is separated into two regions. A smooth dynamical system is defined in each region but the two systems do not match in a continuous way along the discontinuity boundary.

From mid-April until mid-May 2015 I will be at Fudan University as a Senior Visiting Scientist at the Shanghai Key Laboratory of Contemporary Applied Mathematics. There I will be working with Prof. Lin Wei on the dynamics of pulse coupled oscillator networks.

• Vacancy: PhD position
• Topic: Symmetry in classical mechanics
• Affiliation: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science, Dynamical Systems and Mathematical Physics group
• Supervisor: Dr. Konstantinos Efstathiou

Job description

The PhD candidate will study the effect of symmetry on the dynamics and geometry of Hamiltonian systems and work on applications to physical problems. The position includes light teaching duties.

Requirements

Candidates should hold an MSc degree in mathematics, applied mathematics, physics, or related fields.

Since mid-April I am Assistant Professor at the Johann Bernoulli Institute for Mathematics and Computer Science at the University of Groningen.

Our paper with Henk Broer on using covering spaces for proving, and better understanding, fractional monodromy has now been accepted for publication in Communications in Mathematical Physics. Fractional monodromy is a geometric property of integrable Hamiltonian fibrations which generalizes standard Hamiltonian monodromy. In this paper we show that by passing to an appropriate covering space the geometry of the fibration is considerably simplified allowing us to prove fractional monodromy for a very wide class of resonant Hamiltonian systems. For more details you can check the paper.

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.