Current research projects
- Unstable periodic orbits in isotropic turbulence
Collaborators: Shigeo Kida (currently at Doshisha University), Genta Kawahara and Tatsuya Yasuda (School of Engineering Science, Osaka University).
Related publications: V3, VKK, KKV.
Key words: turbulence, periodic orbit theory, scientific computing.
Abstract: We computed unstable periodic solutions in Kida-Pelz flow and investigated their statistical and dynamical properties as well as their dependence on the Reynolds number. The ultimate goal is to compute periodic orbits which reproduce Kolmogorov statistics in the inertial range. Hopefully, this will help us understand the dynamical processes which contribute to turbulent energy dissipation. At this time we are looking into novel ways of computing periodic orbits in very large dissipative systems since existing methods seem to fail at moderate or high Reynolds numbers. We will attempt our computations in homogeneous, isotropic turbulence without any symmetries imposed.
- Global bifurcations in subcritical shear flow
Collaborators: Genta Kawahara.
Related publications: VK, VKM, pV.
Key words: shear turbulence, subcritical transition, global bifurcations, scientific computing.
Abstract: There has been ample interest lately in the problem 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. Such states are called edge states. 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.
- Bifurcation analysis of a mean field cortex model
Collaborators: David Liley, Federico Frascoli (Swinburne University of Technology, Melbourne, Australia);
Ingo Bojak (Radboud University Nijmegen Medical Centre, Netherlands); Loukia Tsakanikas; Kevin Green.
Related publications: VL, FVBL, pFDVBL, bLBDVFF.
Key words: EEG modelling, mean-field model, bifurcation analysis, pattern formation.
Abstract: In this project, we study a mean-field (e.g. PDE) model of human cortical activity. So far, we have performed extensive bifurcation analysis of several low-order reductions of the model. In ongoing work, we are extending this analysis to space and time dependent solutions. 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: in preparation.
Key words: parallel computing, arclength continuation, scientific computing.
Abstract: We are working out a way to parallelize the inherently serial process of arclength continuation, following an elegant, recursive approach. Hopefully, this work will result in a software package that can be applied to dynamical systems-type problems as well as implicit time integration.
- The saddle-node transcritical bifurcation
Collaborators: Ivanky Saputra
Related publications: SQV, SVQ.
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. We have only described a small subset of all possible bifurcation diagrams so far and we would be interested in exploring all possibilities, particularly in the context of mathematical biology.