Normal form of the Bautin bifurcation

We follow the paper^[Kuznetsov] 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:

\[\mathbf L q=i \omega_{0} q, \quad \mathbf L^{T} p=-i \omega_{0} p, \quad \langle p, q\rangle=1.\]

Under some conditions, $x(t)\approx x_0+2\Re w(t)q$ where $w$ satisfies the normal form:

\[\dot{w}=i \omega_{0} w+\frac{1}{2} G_{21} w|w|^{2}+\frac{1}{12} G_{32} w|w|^{4}+O\left(|w|^{6}\right).\tag{E}\]

The second Lyapunov coefficient is

\[l_2:=\frac{1}{12} \operatorname{Re} G_{32}.\]

Normal form computation

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

get_normal_form(br::ContResult, ind_bif::Int ; verbose = false, ζs = nothing, lens = getlens(br))

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 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 Bautin.

References