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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 also have Bifurcations.jl which is unmaintained.

A word on performance

The examples which follow 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.

For example, in one of the simplest tutorials, Temperature model, one could use BandedMatrices.jl for the jacobian and an inplace modification when the jacobian is called.

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 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.

Reproducibility

The documentation of BifurcationKit was built using these direct dependencies,
Status `~/work/BifurcationKitDocs.jl/BifurcationKitDocs.jl/docs/Project.toml`
  [49cd1f3c] AsymptoticNumericalMethod v0.1.0 `https://github.com/bifurcationkit/AsymptoticNumericalMethod.jl#main`
⌅ [aae01518] BandedMatrices v0.17.18
  [0f109fa4] BifurcationKit v0.3.3 `https://github.com/bifurcationkit/BifurcationKit.jl.git#master`
  [52551710] BifurcationKitDocs v0.2.6 `~/work/BifurcationKitDocs.jl/BifurcationKitDocs.jl`
  [9fdde737] DiffEqOperators v4.45.0
⌃ [0c46a032] DifferentialEquations v7.7.0
⌅ [e30172f5] Documenter v0.27.25
⌃ [a106ebf2] DocumenterDiagrams v0.27.0
  [f6369f11] ForwardDiff v0.10.36
⌃ [961ee093] ModelingToolkit v8.50.0
  [d96e819e] Parameters v0.12.3
  [91a5bcdd] Plots v1.40.3
  [295af30f] Revise v3.5.14
⌅ [0bca4576] SciMLBase v1.97.0
  [efcf1570] Setfield v1.1.1
⌅ [47a9eef4] SparseDiffTools v1.31.0
  [37e2e46d] LinearAlgebra
  [2f01184e] SparseArrays v1.10.0
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`
and using this machine and Julia version.
Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 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.
Status `~/work/BifurcationKitDocs.jl/BifurcationKitDocs.jl/docs/Manifest.toml`
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  [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+0
  [3161d3a3] Zstd_jll v1.5.6+0
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.43+1
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.0+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
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 manifest file and the 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