# 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 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 = -0.3, a = -1., b = 1.)
prob = BifurcationProblem(Fbt, [0.01, 0.01], par, (@lens _.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, (@lens _.β2), ContinuationPar(opts_br, detect_bifurcation = 1, max_steps = 40, max_bisection_steps = 25) ;
detect_codim2_bifurcation = 2,
update_minaug_every_step = 1,
bothside = true,
)
# refine BT point
solbt = BifurcationKit.newton_bt(hopf_codim2, 2; start_with_eigen = true)
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.

`BifurcationKit.newton_bt`

— Function```
newton_bt(
prob,
btpointguess,
par,
lens2,
eigenvec,
eigenvec_ad,
options;
normN,
jacobian_ma,
usehessian,
bdlinsolver,
kwargs...
)
```

This function turns an initial guess for a BT point into a solution to the BT problem based on a Minimally Augmented formulation. The arguments are as follows

`prob::AbstractBifurcationFunction`

`btpointguess`

initial guess (x*0, p*0) for the BT point. It should be a`BorderedArray`

as returned by the function`BTPoint`

`par`

parameters used for the vector field`eigenvec`

guess for the 0 eigenvector`eigenvec_ad`

guess for the 0 adjoint eigenvector`options::NewtonPar`

options for the Newton-Krylov algorithm, see`NewtonPar`

.

**Optional arguments:**

`normN = norm`

`bdlinsolver`

bordered linear solver for the constraint equation`jacobian_ma::Symbol = true`

specify the way the (newton) linear system is solved. Can be (:autodiff, :finitedifferences, :minaug)`kwargs`

keywords arguments to be passed to the regular Newton-Krylov solver

**Simplified call**

Simplified call to refine an initial guess for a BT point. More precisely, the call is as follows

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

The parameters / options are as usual except that you have to pass the branch `br`

from the result of a call to `continuation`

with detection of bifurcations enabled and `index`

is the index of bifurcation point in `br`

you want to refine. You can pass newton parameters different from the ones stored in `br`

by using the argument `options`

.

The adjoint of the jacobian `J`

is computed internally when `Jᵗ = nothing`

by using `transpose(J)`

which works fine when `J`

is an `AbstractArray`

. In this case, do not pass the jacobian adjoint like `Jᵗ = (x, p) -> transpose(d_xF(x, p))`

otherwise the jacobian will be computed twice!

For ODE problems, it is more efficient to pass the option `jacobian_ma = :autodiff`

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

This function turns an initial guess for a Bogdanov-Takens point into a solution to the Bogdanov-Takens problem based on a Minimally Augmented formulation.

**Arguments**

`br`

results returned after a call to continuation`ind_bif`

bifurcation index in`br`

**Optional arguments:**

`options::NewtonPar`

, default value`br.contparams.newton_options`

`normN = norm`

`options`

You can pass newton parameters different from the ones stored in`br`

by using this argument`options`

.`jacobian_ma::Symbol = true`

specify the way the (newton) linear system is solved. Can be (:autodiff, :finitedifferences, :minaug)`bdlinsolver`

bordered linear solver for the constraint equation`start_with_eigen = false`

whether to start the Minimally Augmented problem with information from eigen elements.`kwargs`

keywords arguments to be passed to the regular Newton-Krylov solver

For ODE problems, it is more efficient to pass the option `jacobian = :autodiff`

For ODE problems, it is more efficient to pass the option `start_with_eigen = true`

## 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.