Numerical Bifurcation Analysis

The following project proposal is for third-year students taking the mathematical sciences final-year undergraduate project module for the academic year 2026/2027.

Overview and Group Project

This project explores the theoretical foundations and practical applications of numerical bifurcation analysis, an essential technique for understanding how the qualitative behaviour of dynamical systems evolves as system parameters change. The group phase of the project will focus on the theoretical foundations of bifurcation theory. Students will study fixed points and their stability, and learn how to classify bifurcation types , including saddle-node, pitchfork, and Hopf bifurcations, using techniques from dynamical systems theory. Students will also encounter more advanced objects such as stable and unstable manifolds and homoclinic and heteroclinic connections. A key focus will be on numerical techniques for computing bifurcation diagrams, including pseudo-arclength continuation and deflation algorithms.

Bifurcation diagrams of a forest-savanna model from “Pattern formation in Mesic Savannas”, Patterson et al. (2024). The forest tree birth rate (\(\alpha\)) is on the \(x\)-axis in all panels. The resource parameter is denoted by \(r\), and \(G\) is the proportion of the ecosystem occupied by grass. Panels A/B are one-parameter bifurcation diagrams: red curves are stable steady-state solutions, black curves are unstable; blue dots mark transcritical bifurcations and magenta dots mark saddle-node bifurcations. Panel C is a two-parameter bifurcation diagram: blue curves trace transcritical bifurcations and magenta curves trace saddle-node bifurcations.

Techniques

Students will use a combination of analytical and computational methods:

  • Ordinary and partial differential equations and dynamical systems theory
  • Fixed points, stability analysis, and linearisation
  • Classification of bifurcation types (saddle-node, pitchfork, Hopf)
  • Pseudo-arclength continuation and deflation algorithms
  • Numerical continuation software (AUTO, MATCONT, or BifurcationKit.jl)

Mode of Operation and Evidence of Learning

This project is coding-intensive. Students will read and discuss concepts and present their results in weekly meetings. Students will implement numerical methods and use continuation software (AUTO, MATCONT, or BifurcationKit.jl) to generate and analyse bifurcation diagrams for a range of dynamical systems. They will apply dynamical systems theory, including stability analysis, phase plane techniques, and bifurcation classification, both analytically and computationally. Evidence of learning will be demonstrated through a written project report presenting mathematical derivations, computational results, and a discussion of the findings.

Individual Project

The project’s individual phase will focus on the application of these methods to mathematical models from physics and biology, such as predator-prey models, chemical oscillators, and reaction-diffusion equations. Students will use numerical continuation software (AUTO, MATCONT, or BifurcationKit.jl) to generate bifurcation diagrams and investigate how the qualitative behaviour of these models changes as parameters are varied.

Prerequisites

Students should be familiar with basic concepts in differential equations and dynamical systems. This is a coding-intensive project, and prior programming experience is essential. The project is particularly well suited to students who have taken Computational Mathematics II.