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BifurcationKit.jl

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.

Despite initial focus on large scale problems, the package can easily handle low dimensional problems.

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

Hence, it is possible to study large scale nonlinear problems on different hardwares.

One design choice is that we try not to require u to be a subtype of an AbstractArray as this would forbid the use of spectral methods like the one from ApproxFun.jl. For now, our implementation does not allow this for all methods of the package.

Installation

This package requires Julia >= v1.3.0 because of the use of methods added to abstract types (see #31916).

To install it, please run

] add BifurcationKit

To install the bleeding edge version, please run

] add BifurcationKit#master

Citing this work

If you use this package for your work, we ask that you cite the following paper!! Open source development strongly depends on this. 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},
}

Other softwares

There are many good softwares already available.

  • For continuation in small dimension, most softwares are listed on DSWeb. One can mention the widely used AUTO-07p, or also, XPPAUT, MATCONT, PyDSTool and COCO. All these are very reliable and some address high codimensional bifurcations.

  • For large scale problems, there is the versatile and feature full pde2path but also Trilinos-LOCA, CL_MATCONTL, COCO, GetFEM and the python libraries pyNCT and pacopy. There are also some nice lectures by D. Avitabile et al. on matrix free secant continuation based on a concise Matlab implementation which are used in the influential paper [Rankin]. At the time of initial release of BifurcationKit.jl, these lectures provided one of the only libraries for matrix-free continuation, much easier to use than Trilinos-LOCA.

  • For deflated continuation, there is defcont (by the inventor of the algo. P. E. Farrell) and this code by N. M. Evstigneev.

In Julia, we have the following tools:

A word on performance

The tutorials have not all been written with the goal of performance but rather simplicity (except maybe 2d Ginzburg-Landau equation and Langmuir–Blodgett model). One could surely turn them into more efficient codes. The intricacies of PDEs make the writing of efficient code highly problem dependent and one should take advantage of every particularity of the problem under study.

Requested methods for custom arrays

Needless to say, if you use "regular" arrays, you don't need to worry about what follows.

We make the same requirements as KrylovKit.jl. Hence, we refer to its docs and the vector-like interface for more information. We additionally require the following methods to be available:

  • Base.length(x): it is used in the constraint equation of the pseudo arclength continuation method (see continuation for more details). If length is not available for your "vector", define length(x) = 1 and adjust the parameter θ in PALC.
  • Base.copyto!(dest, in) this is required to reduce the allocations
  • Base.eltype must be extended to your vector type.

Citations

The papers citing this work are collected on google scholar.

Some unreferenced papers by google

  • Eydam, Sebastian, Igor Franović, and Louis Kang. “Control of Seizure-like Dynamics in Neuronal Populations with Excitability Adaptation Related to Ketogenic Diet.” Chaos: An Interdisciplinary Journal of Nonlinear Science 34, no. 5 (May 1, 2024): 053128. https://doi.org/10.1063/5.0180954.
  • Stasenko, Sergey V., Sergey M. Olenin, Eugene A. Grines, and Tatiana A. Levanova. “Firing Rate Model for Brain Rhythms Controlled by Astrocytes.” arXiv, May 6, 2024. http://arxiv.org/abs/2405.03601.
  • Olenin, Sergey, Sergey Stasenko, and Tatiana Levanova. “Spiral Attractors in a Reduced Mean-Field Model of Neuron-Glial Interaction.” arXiv, May 7, 2024. http://arxiv.org/abs/2405.04291.

Reproducibility

The documentation of BifurcationKit was built using these direct dependencies,
Status `~/work/BifurcationKitDocs.jl/BifurcationKitDocs.jl/docs/Project.toml`
  [1520ce14] AbstractTrees v0.4.5 `~/.julia/dev/AbstractTrees`
  [49cd1f3c] AsymptoticNumericalMethod v0.2.0
  [aae01518] BandedMatrices v1.8.0 `~/.julia/dev/BandedMatrices`
  [0f109fa4] BifurcationKit v0.4.5
  [52551710] BifurcationKitDocs v0.2.6 `~/work/BifurcationKitDocs.jl/BifurcationKitDocs.jl`
  [0c46a032] DifferentialEquations v7.15.0
  [e30172f5] Documenter v1.8.0
  [a106ebf2] DocumenterDiagrams v1.0.0
  [f6369f11] ForwardDiff v0.10.38
⌅ [961ee093] ModelingToolkit v8.76.0
  [d96e819e] Parameters v0.12.3
  [91a5bcdd] Plots v1.40.9
  [295af30f] Revise v3.6.4
  [0bca4576] SciMLBase v2.67.0
  [efcf1570] Setfield v1.1.1
  [47a9eef4] SparseDiffTools v2.23.0
  [37e2e46d] LinearAlgebra
  [2f01184e] SparseArrays v1.10.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`
and using this machine and Julia version.
Julia Version 1.10.7
Commit 4976d05258e (2024-11-26 15:57 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_DEBUG = Documenter
A more complete overview of all dependencies and their versions is also provided.
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  [7add5ba3] Libgpg_error_jll v1.50.0+0
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⌅ [30392449] Pixman_jll v0.43.4+0
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  [e99dba38] Qt6Wayland_jll v6.7.1+1
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⌅ [fb77eaff] Sundials_jll v5.2.2+0
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  [a3789734] Xorg_libXdmcp_jll v1.1.4+1
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⌅ [dfaa095f] x265_jll v3.5.0+0
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
You can also download the "https://github.com/bifurcationkit/BifurcationKitDocs.jl/tree/gh-pages/v0.2.6/assets/Manifest.toml"">manifest file and the "https://github.com/bifurcationkit/BifurcationKitDocs.jl/tree/gh-pages/v0.2.6/assets/Project.toml"">project file.

References

  • Rankin

    J. Rankin et al., "Continuation of localized coherent structures in nonlocal neural field equations", SIAM J. Scientific Computing 36, pp. B70–B93 (2014): https://epubs.siam.org/doi/10.1137/130918721