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.

Features	Matrix Free	Custom state	Tutorial	GPU
(Deflated) Krylov-Newton	Yes	Yes	link	Yes
Continuation PALC (Natural, Secant, Tangent, Polynomial)	Yes	Yes	All	Yes
Deflated Continuation	Yes	Yes	link	Yes
Bifurcation / Fold / Hopf point detection	Yes	Yes	All / All / link	Yes
Fold Point continuation	Yes	Yes	PDE, PDE, ODE	Yes
Hopf Point continuation	Yes	`AbstractArray`	ODE
Branch point / Fold / Hopf normal form	Yes	Yes		Yes
Branch switching at Branch points	Yes	`AbstractArray`	link	Yes
Automatic bifurcation diagram computation of equilibria	Yes	`AbstractArray`	link
Bogdanov-Takens / Bautin / Cusp / Zero-Hopf / Hopf-Hopf point detection	Yes	Yes	ODE
Bogdanov-Takens / Bautin / Cusp normal forms	Yes	`AbstractArray`	ODE	Yes
Branching from Bogdanov-Takens / Zero-Hopf / Hopf-Hopf to Fold / Hopf curve	Yes	`AbstractArray`	ODE

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).

Features	Method	Matrix Free	Custom state	Tutorial	GPU
Branch switching at Hopf points	SS/PS/OC/T	See each		ODE
Newton / continuation	T	Yes	`AbstractVector`	PDE, PDE	Yes
Newton / continuation	OC		`AbstractVector`	ODE
Newton / continuation	SS	Yes	`AbstractArray`	ODE	Yes
Newton / continuation	PS	Yes	`AbstractArray`	PDE	Yes
Fold, Neimark-Sacker, Period doubling detection	SS/PS/OC/T	See each	`AbstractVector`	link
Branch switching at Branch point	SS/PS/OC/T	See each		ODE
Branch switching at PD point	SS/PS/OC/T	See each		ODE
Continuation of Fold points	SS/PS/OC/T	See each	`AbstractVector`	ODE PDE	Yes
Continuation of Period-doubling points	SS/OC		`AbstractVector`	ODE
Continuation of Neimark-Sacker points	SS/OC		`AbstractVector`	ODE
detection of codim 2 bifurcations of periodic orbits	SS/OC		`AbstractVector`	ODE
Branch switching at Bautin point to curve of Fold of periodic orbits	SS/OC		`AbstractVector`	ODE
Branch switching at ZH/HH point to curve of NS of periodic orbits	SS/OC		`AbstractVector`	ODE

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