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"

	"Parameters used by the vector field."

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

	"Right eigenvectors"

	"Left eigenvectors"

	"Normal form coefficients (basic)"

	"Normal form coefficients (detailed)"

	"Type of bifurcation"

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


  • Haragus

    Haragus, Mariana, and Gérard Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer London, 2011.

  • 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.