Normal form of the Zero-Hopf bifurcation

We follow the paper^[Kuznetsov],^[Kuznetsov2] 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:

\[\mathbf L q_0=0, \quad \mathbf L q_1=i \omega_{0} q_1.\]

Under some conditions, $x(t)\approx x_0+2\Re w_1(t)q_1+w_0(t)q_0$ where $w_i$ satisfy the normal form:

\[\left\{\begin{aligned} \dot{w}_0= & \frac{1}{2} G_{200} w_0^2+G_{011}\left|w_1\right|^2+\frac{1}{6} G_{300} w_0^3 +G_{111} w_0\left|w_1\right|^2+O\left(\left\|\left(w_0, w_1, \bar{w}_1\right)\right\|^4\right) \\ \dot{w}_1= & i \omega_0 w_1+G_{110} w_0 w_1+\frac{1}{2} G_{210} w_0^2 w_1+\frac{1}{2} G_{021} w_1\left|w_1\right|^2 +O\left(\left\|\left(w_0, w_1, \bar{w}_1\right)\right\|^4\right) . \end{aligned}\right.\tag{E}\]

This normal form is usually computed in order to branch from a Zero-Hopf bifurcation point to curves of Neimark-Sacker bifurcations of periodic orbits (see ^[Kuznetsov2]). Not all coefficients in (E) are required for this branching procedure, that is why only a subset of the $G_{ijk}$ is returned.

Normal form computation

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

get_normal_form(br::ContResult, ind_bif::Int ;  
    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 ZeroHopf.

Predictors

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

predictor(zh, , ds; verbose, ampfactor)

predictor(zh, , ds; verbose, ampfactor)

predictor(zh, , ϵ; verbose, ampfactor)

References