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HclinicBifurcationKit.jl

This Julia package aims at performing bifurcation analysis of Homoclinic / Heteroclinic orbits of Cauchy problems.

It builds upon BifurcationKit.jl with version > 0.2 to perform continuation and numerical bifurcation analysis.

Installation

Assuming that you already have Julia correctly installed, it suffices to import HclinicBifurcationKit.jl in the standard way:

import Pkg; Pkg.add("https://github.com/bifurcationkit/HclinicBifurcationKit.jl")

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},
}

Capabilities

  • compute Homoclinic to Hyperbolic Saddle Orbits (HomHS) using Orthogonal collocation or Standard shooting
  • compute bifurcation of HomHS
  • start HomHS from a direct simulation
  • automatic branch switching to HomHS from Bogdanov-Takens bifurcation point

Other softwares

There are several good softwares already available.

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

  • For large scale problems, there is none.

In Julia, the present package seems to be the only one.