I am very happy to supervise highly motivated students on bachelor and master projects in Mathematics, Applied Mathematics, and related disciplines. Please see my research interests and publications and feel free to contact me if something catches your attention. Sometimes I may have specific projects in mind (see below), but some other times we can come up with a project together. Generally speaking, suitable topics include (but are not limited to):
- Slow-fast systems
- Singularity theory
- Bifurcation theory
- Network(ed) Dynamical Systems
- Applications of Dynamical Systems (modeling/analysis/simulation) in Neuroscience, Biochemistry, Epidemiology, Engineering, etc.
Bachelor & Master project-topics readily available:
1. Folded singularities on the cusp catastrophe. (suitable for math & applied math students) [See description]
Folded singularities have been extensively studied in slow-fast systems and appear, for example, as singularities (of the desingularized vector field) sitting exactly on the fold line of a 3-D slow-fast system with 1-fast and 2-slow variables. The objective of the project is to investigate how these singularities interact with other non-hyperbolic points of the critical manifold. In particular, we want to understand what happens to these singularities as they approach a cusp point. [Hide]
2. Interaction between discrete and continuous symmetries. (suitable for math & applied math students) [See description]
In this project we aim to understand better how discrete and continuous symmetries relate in a dynamical system. The idea is to study networks (accounting for discrete symmetries) of a class of equivariant dynamical systems (accounting for continuous symmetries), and to elucidate how they interact when studying bifurcation problems. [Hide]
3. Mathematical modelling of a sPLL (spiking Phase-Locked Loop). (suitable for applied math students) [See description]
A phase-locked loop (PLL) is an input-output system that relates the phase of the output with that of the input in some desired way. There exist several types of PLLs and their applications have been extremely broad. Recently, a new type of PLL has been envisioned: the spiking PLL (sPLL), which plays a fundamental role in the development of innovative Neuromorphic Devices. The aim of the project is to develop a mathematical model, based on ODEs, of an sPLL, which can further be used to aid in the design of neuromorphic devices. This project is in collaboration with the Bio-inspired Circuits and Systems research group.
Desired qualifications: familiarity with ODEs and some experience with modeling, knowledge of a programming language such as Python and/or Matlab, and interest in Neuromorphic Devices. [Hide]
4. Mathematical Analysis of an Economic Model. (suitable for applied math students) [See description]
Recently, a planar economic model with a CES (Constant Elasticity of Substitution) production function has been proposed. This model generalizes economic models that have appeared in the literature for decades. It turns out that this model has two singular properties: a) it has two largely different time scales; and b) the model fails to be Lipschitz at the origin. The objective is to employ geometric techniques of dynamical systems to understand the behavior of such a model. In particular, we aim to elucidate the intricate limit cycles that seem to appear due to the interaction of the aforementioned singular characteristics of the model. [Hide]