2025-10-03 –, Jean-Baptiste Say Amphitheater
Language: English
BifurcationKit.jl is a powerful and flexible Julia package for the numerical study of bifurcations in nonlinear systems. This tutorial will guide participants through the practical use of BifurcationKit.jl to analyze dynamical systems, including fixed points, bifurcation diagrams, continuation of solutions, and codimension-1 bifurcations. Through interactive coding examples, participants will learn how to apply the package to both low- and high-dimensional systems and understand the computational tools behind continuation and stability analysis.
The aim of this tutorial is to introduce the core concepts and workflows of BifurcationKit.jl through hands-on examples, targeting researchers, students, and engineers interested in nonlinear dynamics, pattern formation, and bifurcation theory.
Intended Audience
- Applied mathematicians and physicists working with dynamical systems
- Julia users interested in modeling and simulation
- Graduate students learning bifurcation theory or numerical continuation
- Users of packages like DifferentialEquations.jl or DynamicalSystems.jl
Prerequisites
- Basic knowledge of Julia (functions, packages, plotting)
- Familiarity with differential equations and dynamical systems is helpful but not required
Outline
- Introduction (5 minutes)
- Motivation for bifurcation analysis
- Overview of BifurcationKit.jl capabilities and its ecosystem
- Getting Started (5 minutes)
- Installing and importing BifurcationKit
- Defining a simple bifurcation problem
- Continuation Methods (10 minutes)
- Pseudo-arclength continuation and others
- Tracking solution branches
- Plotting bifurcation diagrams
- Detecting and Analyzing Bifurcations (15 minutes)
- Stability and eigenvalue tracking
- Saddle-node, Hopf, branch point, etc
- Codimension-1 bifurcation examples
- branching from bifurcations (branch / Hopf points)
- Integration with DifferentialEquations.jl and modeling tools
- Focus on periodic orbits (10 minutes)
- Methods for computing them
- Branching from bifurcations
- Codimension-1 bifurcation examples
- Applications and Advanced Topics (15 minutes)
- PDE continuation with spectral methods
- Using Newton-Krylov solvers for large systems
- Q&A and Discussion (5 minutes)
- Troubleshooting tips
- Questions from participants
- Ideas for extensions and open problems
