# 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 function $g_1,g_2$ 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 refinementbelow. In particular, you don't need to set up the Minimally Augmented problem yourself. This is done in the background.

You can pass the bordered linear solver to solve $(M_{bt})$ using the option `bdlinsolver`

(see below). Note that the choice `bdlinsolver = BorderingBLS()`

can lead to singular systems. Indeed, in this case, $(M_{bt})$ is solved by inverting `dF(u,p)`

which is singular at Fold points.

## 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 BifurcationKit, LinearAlgebra, Setfield, SparseArrays, ForwardDiff, Parameters
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 = -0.1, a = -1., b = 1.)
prob = BifurcationProblem(Fbt, [0.01, 0.01], par, (@lens _.β1))
opt_newton = NewtonPar(tol = 1e-9, max_iterations = 40)
opts_br = ContinuationPar(dsmin = 0.001, dsmax = 0.01, ds = 0.01, p_max = 0.5, p_min = -0.5, detect_bifurcation = 3, nev = 2, newton_options = opt_newton, max_steps = 100, n_inversion = 4, dsmin_bisection = 1e-9)
br = continuation(prob, PALC(), opts_br; bothside = true)
# compute branch of Hopf points
hopf_codim2 = continuation(br, 3, (@lens _.β2), ContinuationPar(opts_br, detect_bifurcation = 1, max_steps = 40, max_bisection_steps = 25) ; plot = false, verbosity = 0,
detect_codim2_bifurcation = 2,
update_minaug_every_step = 1,
bothside = true,
bdlinsolver = MatrixBLS(),
)
# refine BT point
solbt = BifurcationKit.newton_bt(hopf_codim2, 2; start_with_eigen = true)
solbt.u
```

```
Bogdanov-Takens bifurcation point at (:β1, :β2) ≈ (-9.030708980674947e-18, 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 Fold / Hopf 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. Reader interested in this advanced usage should look at the code `example/chan.jl`

of the tutorial Temperature model (Simplest example).

## 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.

Missing docstring for `BifurcationKit.newtonBT`

. Check Documenter's build log for details.

## References

- Govaerts
Govaerts, Willy J. F. Numerical Methods for Bifurcations of Dynamical Equilibria. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 2000.

- Blank
Blank, H. J. de, Yu. A. Kuznetsov, M. J. Pekkér, and D. W. M. Veldman. “Degenerate Bogdanov–Takens Bifurcations in a One-Dimensional Transport Model of a Fusion Plasma.” Physica D: Nonlinear Phenomena 331 (September 15, 2016): 13–26. https://doi.org/10.1016/j.physd.2016.05.008.

- Bindel
Bindel, D., M. Friedman, W. Govaerts, J. Hughes, and Yu.A. Kuznetsov. “Numerical Computation of Bifurcations in Large Equilibrium Systems in Matlab.” Journal of Computational and Applied Mathematics 261 (May 2014): 232–48. https://doi.org/10.1016/j.cam.2013.10.034.