# Normal form of the Bogdanov-Takens bifurcation

We follow the book[Haragus] and consider a Cauchy problem

$$$\dot x=\mathbf F(x,p).$$$

We denote by $\mathbf L$ the jacobian of $\mathbf F$ at the bifurcation point $(x_0,p_0)$. We choose a basis such that:

$$$\mathbf{L} \zeta_{0}=0, \quad \mathbf{L} \zeta_{1}=\zeta_{0}.$$$

We can also select a basis:

$$$\mathbf{L}^{*} \zeta_{1}^{*}=0, \quad \mathbf{L}^{*} \zeta_{0}^{*}=\zeta_{1}^{*}$$$

such that

$$$\left\langle\zeta_{0}, \zeta_{0}^{*}\right\rangle=1, \quad\left\langle\zeta_{1}, \zeta_{0}^{*}\right\rangle=0, \quad\left\langle\zeta_{0}, \zeta_{1}^{*}\right\rangle=0, \quad\left\langle\zeta_{1}, \zeta_{1}^{*}\right\rangle=1.$$$

Under some conditions, $x(t)\approx x_0+A(t)\zeta_0 + B(t)\zeta_1$ where $A,B$ satisfy the normal form:

\begin{aligned} &\frac{d A}{d t}=B \\ &\frac{d B}{d t}=\alpha_{1}(\mu)+\alpha_{2}(\mu) A+\alpha_{3}(\mu) B+b A B+a A^{2}\widetilde{\rho}(A, B, \mu) \end{aligned}\tag{E}

where $p = p_0+\mu$ and with coefficients

\begin{aligned} &a=\left\langle\mathbf{F}_{20}\left(\zeta_{0}, \zeta_{0}\right), \zeta_{1}^{*}\right\rangle \\ &b=\left\langle 2 \mathbf{F}_{20}\left(\zeta_{0}, \zeta_{1}\right)-2 \Psi_{200}, \zeta_{1}^{*}\right\rangle. \end{aligned}

The $\Psi$s satisfy

\begin{aligned} a \zeta_{1} &=\mathbf{L} \Psi_{200}+\mathbf{F}_{20}\left(\zeta_{0}, \zeta_{0}\right) \\ b \zeta_{1}+2 \Psi_{200} &=\mathbf{L} \Psi_{110}+2 \mathbf{F}_{20}\left(\zeta_{0}, \zeta_{1}\right) \\ \Psi_{110} &=\mathbf{L} \Psi_{020}+\mathbf{F}_{20}\left(\zeta_{1}, \zeta_{1}\right) \end{aligned}

which gives

$$$0=\left\langle\Psi_{200}, \zeta_{1}^{*}\right\rangle + \left\langle\mathbf{F}_{20}\left(\zeta_{0}, \zeta_{0}\right), \zeta_{0}^{*}\right\rangle.$$$

We conclude that

\begin{aligned} &a=\left\langle\mathbf{F}_{20}\left(\zeta_{0}, \zeta_{0}\right), \zeta_{1}^{*}\right\rangle \\ &b=2\left\langle \mathbf{F}_{20}\left(\zeta_{0}, \zeta_{1}\right), \zeta_{1}^{*}\right\rangle + 2\left\langle\mathbf{F}_{20}\left(\zeta_{0}, \zeta_{0}\right), \zeta_{0}^{*}\right\rangle. \end{aligned}

### Computation of the basis

To build the basis $\left\{\zeta_{0}, \zeta_{1}\right\}$, we follow the procedure described in [AlHdaibat] on page 972.

### Computation of the parameter transform

To invert the mapping $\mu\to (\alpha_{1}(\mu),\alpha_{2}(\mu),\alpha_{3}(\mu))$, we follow the procedure described in [AlHdaibat] on page 956 forward.

## Normal form computation

The normal form (E) can be automatically computed as follows

getNormalForm(br::ContResult, ind_bif::Int ;
nev = 5, verbose = false, ζs = nothing, autodiff = true, detailed = true)

br is a branch computed after a call to continuation with detection of bifurcation points enabled and ind_bif is the index of the bifurcation point on the branch br. The option detailed controls the computation of a simplified version of the normal form. autodiff controls the use of ForwardDiff during the normal form computation.

The above call returns a point with information needed to compute the bifurcated branch. For more information about the optional parameters, we refer to getNormalForm. The result returns the following:

mutable struct BogdanovTakens{Tv, Tpar, Tlens, Tevr, Tevl, Tnf, Tnf2} <: AbstractBifurcationPoint
"Bogdanov-Takens point"
x0::Tv

"Parameters used by the vector field."
params::Tpar

"Parameter axis used to compute the branch on which this BT point was detected."
lens::Tlens

"Right eigenvectors"
ζ::Tevr

"Left eigenvectors"
ζstar::Tevl

"Normal form coefficients (basic)"
nf::Tnf

"Normal form coefficients (detailed)"
nfsupp::Tnf2

"Type of bifurcation"
type::Symbol
end
Note

You should not need to call getNormalForm except if you need the full information about the branch point.

## References

• Haragus

Haragus, Mariana, and Gérard Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer London, 2011. https://doi.org/10.1007/978-0-85729-112-7.

• AlHdaibat

Al-Hdaibat, B., W. Govaerts, Yu. A. Kuznetsov, and H. G. E. Meijer. “Initialization of Homoclinic Solutions near Bogdanov–Takens Points: Lindstedt–Poincaré Compared with Regular Perturbation Method.” SIAM Journal on Applied Dynamical Systems 15, no. 2 (January 2016): 952–80. https://doi.org/10.1137/15M1017491.