Normal form of the Hopf-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:

\[\mathbf L q_1=i \omega_{1} q_1, \quad \mathbf L q_2=i \omega_{2} q_2.\]

Under some conditions, $x(t)\approx x_0+2\Re w_1(t)q_1+2\Re w_2(t)q_2$ where $w_i$ satisfy the normal form:

\[\left\{\begin{aligned} \dot{w}_1= & i \omega_1 w_1+\frac{1}{2} G_{2100} w_1\left|w_1\right|^2+G_{1011} w_1\left|w_2\right|^2 +\frac{1}{12} G_{3200} w_1\left|w_1\right|^4+\frac{1}{2} G_{2111} w_1\left|w_1\right|^2\left|w_2\right|^2+\frac{1}{4} G_{1022} w_1\left|w_2\right|^4 \\ & +O\left(\left\|\left(w_1, \bar{w}_1, w_2, \bar{w}_2\right)\right\|^6\right) \\ \dot{w}_2= & i \omega_2 w_2+G_{1110} w_2\left|w_1\right|^2+\frac{1}{2} G_{0021} w_2\left|w_2\right|^2 +\frac{1}{4} G_{2210} w_2\left|w_1\right|^4+\frac{1}{2} G_{1121} w_2\left|w_1\right|^2\left|w_2\right|^2+\frac{1}{12} G_{0032} w_2\left|w_2\right|^4 \\ & +O\left(\left\|\left(w_1, \bar{w}_1, w_2, \bar{w}_2\right)\right\|^6\right) \end{aligned}\right.\tag{E}\]

This normal form is usually computed in order to branch from a Hopf-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 $G_{ijkl}$ is returned.

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 HopfHopf.

Predictors

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

predictor(hh, , ds; verbose, ampfactor)

predictor(hh, , ϵ; verbose, ampfactor)

References