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 / events located with bisection
  • Compatible with GPU
  • 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, Cusp, Zero-Hopf. (Hopf-Hopf normal form not implemented)
  • Branching from Bogdanov-Takens / Zero-Hopf / Hopf-Hopf points to Fold / Hopf curve
  • continuation of fixed points of maps
  • computation of normal form of Period-doubling, Neimark-Sacker, Branch point bifurcations.

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

Custom state means, you 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 continuationYesYesPDE, PDE, 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
  • PO computation and continuation using parallel (Standard or Poincaré) Shooting, Finite Differences or Orthogonal Collocation (mesh adaptive).
  • Automatic branch switching from simple Hopf points to PO
  • Automatic branch switching from simple Period-Doubling points to PO
  • Assisted branch switching from simple Branch points to PO
  • Detection of Branch, Fold, Neimark-Sacker (NS), Period Doubling (PD) bifurcation points of PO.
  • Fold / PD / NS continuation based on Minimally Augmented formulation (for shooting and collocation). Trapezoid method only allows continuing Fold of PO.
  • Detection of all codim 2 bifurcations of PO (R1, R2, R3, R4, GPD, NS-NS, Chenciner, Fold-Flip, Fold-NS, PD-NS)
  • Computation of the normal forms of PD, NS (for shooting and collocation) using the method based on Poincaré return map or the Iooss normal form (more precise).
  • automatic branching from Bautin to curve of Fold of PO
  • automatic branching from Zero-Hopf to curve of NS of PO
  • automatic branching from Hopf-Hopf to curve of NS of PO

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 eachODE
Newton / continuationTYesAbstractVectorPDE, PDEYes
Newton / continuationOCAbstractVectorODE
Newton / continuationSSYesAbstractArrayODEYes
Newton / continuationPSYesAbstractArrayPDEYes
Fold, Neimark-Sacker, Period doubling detectionSS/PS/OC/TSee eachAbstractVectorlink
Branch switching at Branch pointSS/PS/OC/TSee eachODE
Branch switching at PD pointSS/PS/OC/TSee eachODE
Continuation of Fold pointsSS/PS/OC/TSee eachAbstractVectorODE PDEYes
Continuation of Period-doubling pointsSS/OCAbstractVectorODE
Continuation of Neimark-Sacker pointsSS/OCAbstractVectorODE
detection of codim 2 bifurcations of periodic orbitsSS/OCAbstractVectorODE
Branch switching at Bautin point to curve of Fold of periodic orbitsSS/OCAbstractVectorODE
Branch switching at ZH/HH point to curve of NS of periodic orbitsSS/OCAbstractVectorODE

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

List of detected bifurcations

A left-to-right arrow in the following graph from $E_1$ to $E_2$ means that $E_2$ can be detected when continuing an object of type $E_1$.

A right-to-left arrow from $E_2$ to $E_1$ means that we can start the computation of object of type $E_1$ from $E_2$.

Each object of codim 0 (resp. 1) can be continued with 1 (resp. 2) parameters.

graph LR S[ ] C[ Equilibrium ] PO[ Periodic orbit ] BP[ Fold/simple branch point ] H[ Hopf \n :hopf] CP[Cusp] BT[ Bogdanov-Takens \n :bt ] ZH[Zero-Hopf \n :zh] GH[Bautin \n :gh] HH[Hopf-Hopf \n :hh] FPO[ Fold Periodic orbit ] NS[ Neimark-Sacker \n :ns] PD[ Period Doubling \n :pd ] BPC[BPC] CH[Chenciner \n :ch] GPD[Generalized period doubling \n :gpd] BPC[Branch point PO] LPPD[Fold-Flip] LPNS[Fold-NeimarkSacker] R1[1:1 resonance point\n :R1] R2[1:2 resonance point\n :R2] R3[1:3 resonance point\n :R3] R4[1:4 resonance point\n :R4] S --> C S --> PO C --> nBP[ non simple\n branch point ] C --> BP C --> H BP --> CP BP <--> BT PO --> H PO --> FPO PO --> NS PO --> PD FPO <--> GH FPO <--> BPC FPO --> R1 NS --> R1 NS --> R3 NS --> R4 NS --> CH NS --> LPNS NS --> NSNS NS --> R2 NS --> PDNS PD --> PDNS PD --> R2 PD --> LPPD PD --> GPD H <--> BT H <--> ZH BP <--> ZH H <--> HH H <--> GH NS <--> ZH PO <--> BPC NS <--> HH FPO --> LPNS FPO --> LPPD _ --> Codim0 --> Codim1 --> Codim2