Past research projects
Collaborators: Genta Kawahara, Susumu Goto (Osaka University, Japan); Behzad Nikzad.
Related publications: VK, VKM, pV, VGo.
Key words: shear turbulence, subcritical transition, global bifurcations, scientific computing.
Abstract: Since the seminar work by Nagata in the early 1980s, there has been ample interest in the mathematical description of subcritical transition to turbulence in shear flows. The edge state hypothesis asserts that certain flows with a simple spatial structure mediate between the laminar and turbulent states of shear flow. We are trying to unravel the global connections (homoclinic or heteroclinic) of such states, going on the idea that subcritical turbulence is a random-ish walk between many, possibly infinitely many, invariant solutions such as equilibria, periodic orbits and traveling waves. To this end we use qualitatively correct low-order models as well as full-fledged direct numerical simulations.
Dynamical analysis of mean field cortex models
Collaborators: David Liley, Federico Frascoli (Swinburne University of Technology, Melbourne, Australia);
Ingo Bojak (University of Reading, UK); Loukia Tsakanikas, Kevin Green, Laura Green.
Related publications: VL, FVBL, pFDVBL, bLBDVFF, GV, VG.
Key words: EEG modelling, mean-field model, bifurcation analysis, pattern formation.
Abstract: In this project, we study a mean-field (PDE) model of human cortical activity. We have performed extensive simulations and bifurcation analysis of the model and several low-order reductions. Some of the questions we address concern the generation of the alpha rhythm, the effect of anaesthetic agents and the effect of external noise.
A parallel method for pseudo-arclength continuation
Collaborators: Dhavide Aruliah and Alex Dubitski.
Related publications: AVD, PAVD.
Key words: parallel computing, arclength continuation, scientific computing.
Abstract: We worked out a way to parallelize the inherently serial process of arclength continuation, following an elegant, recursive approach. This work resulted in a software package that can be applied to computationally intense continuation problems.
The saddle-node transcritical bifurcation
Collaborators: Ivanky Saputra, Reinout Quispel, Marvin Hoti
Related publications: SQV, SVQ, VH.
Key words: saddle-node-transcritical bifurcation, normal form theory, mathematical biology.
Abstract: This project grew out of Ivanky's PhD thesis and focuses on the interaction between saddle-node and transcritical bifurcations. The dynamics generated by this interaction are surprisingly rich, in fact they can be found in the unfolding of a degenerate Bogdanov-Takens point. The papers describe the unfolding of this singularity and its detection in a model of predator-prey-toxicant interaction.
1997-2002 Ph.D. at the mathematical institute of Utrecht university and the Royal Dutch Meteorological Institute, KNMI, supervised by Ferdinand Verhulst and Theo Opsteegh. The results are summarised in my thesis, "Time scale interaction in low-order climate models".
I used bifurcation and continuation techniques to analyse low-order models of the atmosphere and the ocean. This resulted in a mix of modeling, scaling, Galerkin truncation, bifurcation analysis, singular perturbations and a considerable amount of trouble to relate all that to "reality".
Another paper (KMV), not included in my thesis, is the result of joint work with Yuri Kuznetsov and Hil Meijer on a certain codimension two bifurcation.
ITFA, institute for theoretical physics of the university of Amsterdam. I specialized in statistical physics, writing my thesis on quasi-crystalline structure and random tiling models. The project was supervised by Bernard Nienhuis and Jan de Gier.
For a simple introduction to quasi crystals and random tiling models you might leaf through my thesis.