BifurcationKit.jl Documentation

BifurcationKit.jl is designed for automatic bifurcation analysis of large-dimensional equations $F(u, λ)=0$ (where $λ$ is real).

Key features include:

• Continuation Algorithms: Supports PALC, deflated continuation, and others, using Newton-Krylov methods for predictor step correction.
• Stability Analysis: Uses Matrix-Free, Dense, or Sparse eigensolvers to compute stability and bifurcation points.
• Periodic Orbits: Provides both shooting methods and methods based on finite differences/collocation to compute periodic orbits of Cauchy problems.
• Scalability: Optimized for large-scale nonlinear problems, supporting Matrix-Free methods on GPUs and on clusters for studying nonlinear PDEs and nonlocal problems.

BifurcationKit.jl is integrated into the SciML ecosystem. It is used as an external dependency by:

• ModelingToolkit.jl
• Catalyst.jl

It is also listed under the 'Analysis' menu in the SciML documentation for third-party parameter analysis.

BifurcationKit.jl requires Julia version >= v1.3.0. You can install the stable version or the bleeding edge (master) version using the Julia package manager.

# Install the stable version
using Pkg
Pkg.add("BifurcationKit")

# OR install the bleeding edge version
Pkg.add(url="https://github.com/rveltz/BifurcationKit.jl", rev="master")

The ecosystem is extended by several specialized plugins located under the bifurcationkit organization:

• HclinicBifurcationKit.jl: Bifurcation analysis of homoclinic/heteroclinic orbits of ODEs.
• DDEBifurcationKit.jl: Bifurcation analysis of delay differential equations (DDE).
• AsymptoticNumericalMethod.jl: Provides the Asymptotic Numerical Method (ANM) algorithm for use within BifurcationKit.jl.
• GridapBifurcationKit.jl: Bifurcation analysis of PDEs solved with the Finite Elements Method (FEM) using Gridap.jl.
• PeriodicSchurBifurcationKit.jl: Computes Floquet coefficients for the stability analysis of periodic orbits.