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

It incorporates continuation algorithms (PALC, deflated continuation, ...) based on a Newton-Krylov method to correct the predictor step and a Matrix-Free/Dense/Sparse eigensolver is used to compute stability and bifurcation points.

The idea is to be able to seamlessly switch the continuation algorithm a bit like changing the time stepper (Euler, RK4,...) for ODEs.

BifurcationKit can also seek for periodic orbits of Cauchy problems. It is by now, one of the only software which provides shooting methods and methods based on finite differences / collocation to compute periodic orbits.

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

Despite this focus, the package can easily handle low dimensional problems and specific optimizations are regularly added.

If you use BifurcationKit.jl in 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},
}

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

Most plugins are located in the organization bifurcationkit:

• HclinicBifurcationKit.jl bifurcation analysis of homoclinic / heteroclinic orbits of ordinary differential equations (ODE)
• DDEBifurcationKit.jl bifurcation analysis of delay differential equations (DDE)
• AsymptoticNumericalMethod.jl provides the numerical continuation algorithm Asymptotic Numerical Method (ANM) which can be used directly in BifurcationKit.jl
• GridapBifurcationKit.jl bifurcation analysis of PDEs solved with the Finite Elements Method (FEM) using the package Gridap.jl.
• PeriodicSchurBifurcationKit.jl state of the art computation of Floquet coefficients, useful for computing the stability of periodic orbits.

The list of capabilities is available here.

simple ODE example	Codimension 2 (ODE)
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