Overview of capabilities

• Overview of capabilities
• • Main features
  • Capabilities related to equilibria
  • (limited) Capabilities related to maps
  • Capabilities related to Periodic orbits (PO)
  • Capabilities related to Homoclinic orbits
  • List of detected bifurcations

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

Capabilities related to equilibria

• 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

(limited) Capabilities related to maps

• 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.

Capabilities related to Periodic orbits (PO)

• 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).

Capabilities related to Homoclinic orbits

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