kristofferc / bifurcationkit.jl

A Julia package to perform Bifurcation Analysis

BifurcationKit.jl

Documentation	Build Status

This Julia package aims at performing automatic bifurcation analysis of large dimensional equations F(u, λ)=0 where λ∈ℝ by taking advantage of iterative methods, sparse formulation and specific hardwares (e.g. GPU).

It incorporates continuation algorithms (PALC, deflated continuation, ...) which provide a predictor (u1, λ1) from a known solution (u0, λ0). A Newton-Krylov method is then used to correct this predictor and a Matrix-Free eigensolver is used to compute stability and bifurcation points.

By leveraging on the above method, it can also seek for periodic orbits of Cauchy problems by casting them into an equation F(u, λ)=0 of high dimension. It is by now, one of the only softwares which provides shooting methods AND methods based on finite differences to compute periodic orbits.

The current package focuses on large scale nonlinear problems and multiple hardwares. Hence, the goal is to use Matrix Free methods on GPU (see PDE example and Periodic orbit example) or on a cluster to solve non linear PDE, nonlocal problems, compute sub-manifolds...

Support and citation

If you use this package for your work, we ask that you cite the following paper. Open source development as part of academic research strongly depends on this. Please also consider starring this repository if you like our work, this will help us to secure funding in the future. It is referenced on HAL-Inria as follows:

@misc{veltz:hal-02902346,
  TITLE = {{BifurcationKit.jl}},
  AUTHOR = {Veltz, Romain},
  URL = {https://hal.archives-ouvertes.fr/hal-02902346},
  INSTITUTION = {{Inria Sophia-Antipolis}},
  YEAR = {2020},
  MONTH = Jul,
  KEYWORDS = {pseudo-arclength-continuation ; periodic-orbits ; floquet ; gpu ; bifurcation-diagram ; deflation ; newton-krylov},
  PDF = {https://hal.archives-ouvertes.fr/hal-02902346/file/354c9fb0d148262405609eed2cb7927818706f1f.tar.gz},
  HAL_ID = {hal-02902346},
  HAL_VERSION = {v1},
}

Installation

This package requires Julia >= v1.3.0

To install it, please run

] add BifurcationKit

To install the bleeding edge version, please run

] add BifurcationKit#master

Website

The package is located here.

Examples of bifurcation diagrams

Automatic Bif. Diagram in 1D Swift Hohenberg	Automatic Bif. Diagram in 2D Bratu
Snaking in 2D Swift Hohenberg	Periodic orbits in 1D Brusselator
Period doubling BVAM Model	Periodic orbits in 2D Ginzburg-Landau
Deflated Continuation in Carrier problem	2D Swift Hohenberg on GPU

Main features

• Newton-Krylov solver with generic linear / eigen preconditioned solver. Idem for the arc-length continuation.
• Continuation written as an iterator
• Monitoring user functions along curves computed by continuation, see events
• Newton-Krylov solver with nonlinear deflation and preconditioner. It can be used for branch switching for example.
• Deflated continuation
• Bifurcation points are located using a bisection algorithm
• Branch, Fold, Hopf bifurcation point detection of stationary solutions.
• Automatic branch switching at branch points (whatever the dimension of the kernel)
• Automatic branch switching at simple Hopf points to periodic orbits
• Automatic bifurcation diagram computation
• Fold / Hopf continuation based on Minimally Augmented formulation, with Matrix Free / Sparse Jacobian.
• detection of Bogdanov-Takens, Bautin and Cusp bifurcations
• Periodic orbit computation and continuation using Shooting or Finite Differences.
• Branch, Fold, Neimark-Sacker, Period Doubling bifurcation point detection of periodic orbits.
• Computation and Continuation of Fold of periodic orbits

Custom state means, we can use something else than AbstractArray, for example your own struct.

Note that you can combine most of the solvers, like use Deflation for Periodic orbit computation or Fold of periodic orbits family.

Features	Matrix Free	Custom state	Tutorial	GPU
(Deflated) Krylov-Newton	Yes	Yes	All	✔️
Continuation (Natural, Secant, Tangent, Polynomial)	Yes	Yes	All	✔️
Deflated Continuation	Yes	Yes	⤴️	✔️
Branching / Fold / Hopf point detection	Yes	Yes	All / All / ⤴️	✔️
Fold Point continuation	Yes	Yes	⤴️, ⤴️	✔️
Hopf continuation	Yes	AbstractArray	⤴️
Branch switching at Branch / Hopf points	Yes	AbstractArray	⤴️	✔️
Automatic bifurcation diagram computation	Yes	AbstractArray	⤴️
Periodic Orbit (FD) Newton / continuation	Yes	AbstractVector	⤴️, ⤴️	✔️
Periodic Orbit with Parallel Poincaré / Standard Shooting Newton / continuation	Yes	AbstractArray	⤴️
Fold, Neimark-Sacker, Period doubling detection	Yes	AbstractVector	⤴️
Continuation of Fold of periodic orbits	Yes	AbstractVector	⤴️	✔️

To do or grab

Without a priority order:

• improve compatibility with DifferentialEquations.jl
• Add interface to other iterative linear solvers (cg, minres,...) from IterativeSolvers.jl