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

get_normal_form(br::ContResult, ind_bif::Int;
    ζs = nothing, 
    autodiff = true, 
    detailed = true,
    kwargs...)

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 get_normal_form. The result returns an object of type BogdanovTakens.

Predictors

The predictor for a non trivial guess at distance $\delta p$ from the bifurcation point is provided by the method

predictor(bt, , ds; verbose, ampfactor)

predictor(bt, , ds; verbose, ampfactor)

predictor(bt, , ds; verbose, ampfactor)

References