# Research

My research deals principally with ordinary differential equations systems and stability.

## Pattern Formation on a Spherical Cap

My current research deals with reaction-diffusion patterns on a spherical cap with slowly changing curvature.
This is motivated by plant embryo growth where cotyledons (embryonic leaves) usually grow in a pattern similar to harmonic functions at the same time that the embryonic tip flattens.
It has been previously shown (Nagata, Zangeneh, Holloway, 2013) that the transition from a patternless state to a single-mode patterned state in a Brusselator system can be described by a pitchfork or transcritical bifurcation in a quasi-static approximation, where we assume that the system reaches equilibrium upon changing the parameter values.
We have an example of a two ringed and three periods harmonic mode in the figure below, obtained from a finite elements simulation of the Brusselator.

Example of a (3,2) mode, with two rings and three periods per ring

Example of a mixed pattern with the (5,1) and (0,3) modes

We extended the quasi-static predictions to areas in parameter space where two modes may be in competition or coexistence. This led to codimension 2 bifurcations of the double pitchfork or transcritical-pitchfork type. The latter case, combining a ringed and a spotted pattern, was interesting as it involved some small terms, which we included in the computation of some higher order terms and led to a complex bifurcation diagram, with many stability cases.
We wish to use bifurcation theory to efficiently predict pattern emergence on such a system, initially with constant curvature, then by introducing a slowly decreasing curvature with time. In the latter case we observed that the same patterns persist through what is called a delayed bifurcation.

## Numerical Methods

The results we obtain from the ordinary differential equations have to be compared to numerical simulations of the Brusselator system in order to validate them.
We initially used finite elements methods to produce such simulations, such as the two pictures above and we are trying to adapt the
closest point method,
a new embedding method for partial differential equations.

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