Piecewise-smooth dynamical systems

About the Project

Mathematical models of systems that switch between different types of evolution suffer a curse of dimensionality. Their dynamical behaviour cannot be faithfully reproduced with fewer equations. Researchers currently persist with low-dimensional models, or suffice with numerical computations and observations. The PhD research will be part of a Marsden funded project to develop new strategies for resolving this issue by exploiting the poorly recognised but common occurrence of degeneracies in high-dimensional models. The degeneracies arise for diverse reasons, including energy conservation, optimisation, and physical constraints, and are often obscured until the model is viewed abstractly. The over-arching idea is that the degeneracies provide a novel and over-looked avenue by which dimensionality can be reduced and the models can be analysed thoroughly.

Two scholarships are available. Each scholarship covers all tuition fees for international and domestic students and includes a tax-free stipend of NZ$35,000 annually for up to three years.

The students will work with difference equations and ordinary differential equations under the supervision of Assoc. Prof. David Simpson. They will develop new theoretical results when discontinuities are present in the equations through asymptotic methods supported by numerical simulations. The students will apply the results to box models of ocean circulation and/or to the internal dynamics of deep neural networks central to the functioning of modern artificial intelligence.

To apply please email David a CV, academic transcript, and cover letter.

Funding Notes

Royal Society of New Zealand (Marsden grant 25-MAU-044).