Journal

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.

On 31 January and 1 February 2011, Henk Broer, Xia Liu, and I, organize a workshop with subject Resonance and Synchronization. The purpose of the workshop is to explore the interaction between these two dynamical concepts and their applications in biological and mechanical systems. More information about the workshop including information about registration and about the program is available on the workshop’s website .

Between 14 and 20 January 2011 I am visiting Andrea Giacobbe and Francesco Fassò at the Mathematics department in Padova to continue the work we did during my previous visit.

In 2010 I continued working on research problems on dynamical systems and the geometry of integrable Hamiltonian fibrations. Here is a short list of 2010 events.

• Four papers, finished in the previous year, were published in 2010. Two more papers, one on a covering spaces approach to fractional Hamiltonian monodromy and the other on circadian rhythms, were finished and are now under review.

• Started a collaboration with Andrea Giacobbe which in 2010 brought Andrea once to Groningen and me once to Padova.

Teaching period is over. During this period I was invited to give two talks but I didn´t have until now the opportunity to write about these here. In both occasions I talked about my favorite subject at the moment: integrable Hamiltonian systems with codimension-1 singularities.

Firstly, I was invited to give a talk at the second meeting of the GDR Quantum Dynamics that took place 24-26 March 2010 in Dijon. Besides my presentation, I had the opportunity to discuss with my colleagues in Dijon (Dominique Sugny, Pavao Mardešić, Michèle Pelletier, and Hans Jauslin) and with Dmitrií Sadovskií.

Three new papers have been accepted for publication recently. First, a review paper with Dmitrií Sadovskií has been accepted for publication in Reviews of Modern Physics. The subject of the review is the recent advances in the study of the hydrogen atom in electric and magnetic fields that were made possible by adopting the point of view of global analysis of (near) integrable Hamiltonian systems. A paper on bidromy with Dominique Sugny has been already published in Journal of Physics A. In the paper we describe the classical actions and the joint quantum spectrum of integrable Hamiltonian systems with bifurcation diagram that contains a swallowtail catastrophe. Finally, a paper with Elie Assémat, Marc Joyeux, and Dominique Sugny has been accepted for publication to Physical Review Letters. In this paper we study the common appearance of fractional monodromy and bidromy in the vibrational spectrum of HOCl.

I´m just back from my vacation trip to the very beautiful Iceland, so the new year (in terms of work) starts a little late. I have realized that time plays very strange tricks with my memories and as I grow older things become worse. In particular, when I look back I tend to underestimate the things I have achieved in any given time period and how much I have developed as a researcher and as a person. Writing some of these things down is the best solution to keep things into perspective. So, I am starting something that I will try to make a tradition in this journal: a review of the past year.

The main requirement for numerical code is that it must be fast. This means that numerical code is usually written in some “low-level” language such as C or Fortran. Such code can be highly optimized but low-level languages are not very expressive (where expressiveness is defined as the inverse of the number of lines of code that is necessary in order to express a program). Thus they are not very well suited to other tasks, such as data I/O, command line argument parsing, graphics, and user interaction that are peripheral to, but nevertheless important for, the computation and where speed does not play the primary role.

I have been unhappy for some time now with the function that I had written in Mathematica for computing the normal form of a Hamiltonian system. I used the standard Lie series approach but I had written the code using a traditional approach with several Do loops. Except for the fact that this kind of code is completely contrary to Mathematica´s philosophy I found the code hard to read and inefficient to the degree at least that Mathematica can be efficient.