Bogdanov-Takens refinement

In this page, we explain how to perform precise localisation of Bogdanov-Takens (BT) points. This is an unusual feature of numerical continuation libraries. We chose to implement it because the localisation of the BT points on the Hopf bifurcation curves is rather imprecise.

Method

The continuation of BT bifurcation points is based on a Minimally Augmented^[Govaerts],^[Blank],^[Bindel] formulation which is an efficient way to detect singularities. The continuation of BT points is based on the formulation

\[G(u,p) = (F(u,p), g_1(u,p), g_2(u,p))\in\mathbb R^{n+2}\quad\quad (F_{bt})\]

where the test functions $g_1,g_2$ are solutions of

\[\left[\begin{array}{cc} dF(u,p) & w \\ v^{\top} & 0 \end{array}\right]\left[\begin{array}{c} v_1 \\ g_1(u,p) \end{array}\right]=\left[\begin{array}{c}0_{n} \\1\end{array}\right]\quad\quad (M_{bt})\]

and

\[\left[\begin{array}{cc} dF(u,p) & w \\ v^{\top} & 0 \end{array}\right]\left[\begin{array}{c} v_2 \\ g_2(u,p) \end{array}\right]=\left[\begin{array}{c}v_1 \\0\end{array}\right]\quad\quad (M_{bt})\]

and where $w,v$ are chosen in order to have a non-singular matrix $(M_{bt})$. More precisely, $v$ (resp. $w$) should be close to a null vector of dF(u,p) (resp. dF(u,p)').

note that there are very simplified calls for this, see Newton refinement below. In particular, you don't need to set up the Minimally Augmented problem yourself. This is done in the background.

Setting the jacobian

In order to apply the newton algorithm to $F_{bt}$, one needs to invert the jacobian. This is not completely trivial as one must compute this jacobian and then invert it. You can select the following jacobians for your computations (see below):

• [Default] for jacobian_ma = :autodiff, automatic differentiation is applied to $F_{bt}$ and the matrix is then inverted using the provided linear solver. In particular, the jacobian is formed. This is very well suited for small dimensions (say < 100)
• for jacobian_ma = :minaug, a specific procedure for evaluating the jacobian $F_{bt}$ and inverting it (without forming the jacobian!) is used. This is well suited for large dimensions.

Example

using Revise, BifurcationKit
Fbt(x, p) = [x[2], p.β1 + p.β2 * x[2] + p.a * x[1]^2 + p.b * x[1] * x[2]]
par = (β1 = 0.01, β2 = -0.3, a = -1., b = 1.)
prob  = BifurcationProblem(Fbt, [0.01, 0.01], par, (@optic _.β1))
opts_br = ContinuationPar(p_max = 0.5, p_min = -0.5, detect_bifurcation = 3, nev = 2)

br = continuation(prob, PALC(), opts_br; bothside = true)

# compute branch of Hopf points
hopf_codim2 = continuation(br, 3, (@optic _.β2), ContinuationPar(opts_br, max_steps = 40) ;
	detect_codim2_bifurcation = 2,
	update_minaug_every_step = 1,
	bothside = true,
	)

# refine BT point
solbt = BifurcationKit.newton_bt(hopf_codim2, 2)
solbt.u

Bogdanov-Takens bifurcation point at (:β1, :β2) ≈ (0.0, 0.0).
Normal form (B, p1 + p2⋅B + b⋅A⋅B + a⋅A²)
Normal form coefficients:
 a = missing
 b = missing

You can call various predictors:
 - predictor(::BogdanovTakens, ::Val{:HopfCurve}, ds)
 - predictor(::BogdanovTakens, ::Val{:FoldCurve}, ds)
 - predictor(::BogdanovTakens, ::Val{:HomoclinicCurve}, ds)

Newton refinement

Once a Bogdanov-Takens point has been detected after a call to br = continuation(...), it can be refined using newton iterations. Let us say that ind_bif is the index in br.specialpoint of a Bogdanov-Takens point. This guess can be refined as follows:

outfold = newton(br::AbstractBranchResult, ind_bif::Int;  
	normN = norm,
	options = br.contparams.newton_options,
	bdlinsolver = BorderingBLS(options.linsolver),
	jacobian_ma = :autodiff,
	start_with_eigen = false, kwargs...)

For the options parameters, we refer to Newton.

It is important to note that for improved performances, a function implementing the expression of the hessian should be provided. This is by far the fastest. BifurcationProblem provides it by default using AD though.

Advanced use

Here, we expose the solvers that are used to perform newton refinement. This is useful in case it is too involved to expose the linear solver options.

newton_bt(
    prob,
    btpointguess,
    par,
    lens2,
    eigenvec,
    eigenvec_ad,
    options;
    normN,
    jacobian_ma,
    usehessian,
    bdlinsolver,
    bdlinsolver_adjoint,
    bdlinsolver_block,
    kwargs...
)

newton(br::AbstractBranchResult, ind_bt::Int; options = br.contparams.newton_options, kwargs...)

newton_bt(
    br,
    ind_bt;
    probvf,
    normN,
    options,
    nev,
    start_with_eigen,
    bdlinsolver,
    bdlinsolver_adjoint,
    kwargs...
)

References