Overview of capabilities

Main features

  • Newton-Krylov solver with generic linear / eigen preconditioned solver. Idem for the arc-length continuation.
  • Newton-Krylov solver with nonlinear deflation and preconditioner. It can be used for branch switching for example. It is used for deflated continuation.
  • Continuation written as an iterator
  • Monitoring user functions along curves computed by continuation, see events
  • Continuation methods: PALC, Moore Penrose, Multiple, Polynomial, Deflated continuation, ANM, ...
  • Bifurcation points located with bisection
  • Detection of Branch, Fold, Hopf bifurcation points of stationary solutions and computation of their normal form.
  • Automatic branch switching at branch points (whatever the dimension of the kernel) to equilibria
  • Automatic computation of bifurcation diagrams of equilibria
  • Fold / Hopf continuation based on Minimally Augmented formulation, with Matrix Free / Sparse / Dense Jacobian.
  • Detection of all codim 2 bifurcations of equilibria and computation of the normal forms of Bogdanov-Takens, Bautin and Cusp
  • Branching from Bogdanov-Takens / Zero-Hopf / Hopf-Hopf points to Fold / Hopf curve

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

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

FeaturesMatrix FreeCustom stateTutorialGPU
(Deflated) Krylov-NewtonYesYeslinkYes
Continuation PALC (Natural, Secant, Tangent, Polynomial)YesYesAllYes
Deflated ContinuationYesYeslinkYes
Bifurcation / Fold / Hopf point detectionYesYesAll / All / linkYes
Fold Point continuationYesYeslink, link, ODEYes
Hopf Point continuationYesAbstractArrayODE
Branch point / Fold / Hopf normal formYesYesYes
Branch switching at Branch pointsYesAbstractArraylinkYes
Automatic bifurcation diagram computation of equilibriaYesAbstractArraylink
Bogdanov-Takens / Bautin / Cusp / Zero-Hopf / Hopf-Hopf point detectionYesYesODE
Bogdanov-Takens / Bautin / Cusp normal formsYesAbstractArrayODEYes
Branching from Bogdanov-Takens / Zero-Hopf / Hopf-Hopf to Fold / Hopf curveYesAbstractArrayODE
  • Periodic orbit computation and continuation using parallel (Standard or Poincaré) Shooting, Finite Differences or Orthogonal Collocation.
  • Automatic branch switching at simple Hopf points to periodic orbits
  • Detection of Branch, Fold, Neimark-Sacker, Period Doubling bifurcation points of periodic orbits.
  • Continuation of Fold of periodic orbits

Legend for the table: Standard shooting (SS), Poincaré shooting (PS), Orthogonal collocation (OC), trapezoid (T).

FeaturesMethodMatrix FreeCustom stateTutorialGPU
Branch switching at Hopf pointsSS/PS/OC/TSee eachlink
Newton / continuationTYesAbstractVectorlink, linkYes
Newton / continuationOCAbstractVectorlink
Newton / continuationSSYesAbstractArraylinkYes
Newton / continuationPSYesAbstractArraylinkYes
Fold, Neimark-Sacker, Period doubling detectionSS/PS/OC/TSee eachAbstractVectorlink
Branch switching at Branch pointSS/PS/OC/TSee eachlink
Branch switching at PD pointSS/PS/OC/TSee eachlink
Continuation of Fold pointsSS/PS/OC/TSee eachAbstractVectorlinkYes
Continuation of Period-doubling pointsSS/OCAbstractVectorlink
Continuation of Neimark-Sacker pointsSS/OCAbstractVectorlink
detection of codim 2 bifurcations of periodic orbitsSS/OCAbstractVectorlink
Branch switching at Bautin point to curve of Fold of periodic orbitsSS/OCAbstractVector
Branch switching at ZH/HH point to curve of NS of periodic orbitsSS/OCAbstractVector

This is available through the plugin HclinicBifurcationKit.jl. Please see the specific docs for more information.

  • 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-Takes bifurcation point