Simple Hopf point

At a Hopf branch point $(x_0,p_0)$ for the problem $F(x,p)=0$, the spectrum of the linear operator $dF(x_0,p_0)$ contains two purely imaginary $\pm i\omega,\ \omega > 0$ which are simple. At such point, we can compute the normal form to transform the initial Cauchy problem

$$$\dot x = \mathbf{F}(x,p)$$$

in large dimensions to a complex polynomial vector field ($\delta p\equiv p-p_0$):

$$$\dot z = z\left(a \cdot\delta p + i\omega + l_1|z|^2\right)\quad\text{(E)}$$$

whose solutions give access to the solutions of the Cauchy problem in a neighborhood of $(x,p)$.

More precisely, if $\mathbf{J} \equiv d\mathbf{F}(x_0,p_0)$, then we have $\mathbf{J}\zeta = i\omega\zeta$ and $\mathbf{J}\bar\zeta = -i\omega\bar\zeta$ for some complex eigenvector $\zeta$. It can be shown that $x(t) \approx x_0 + 2\Re(z(t)\zeta)$ when $p=p_0+\delta p$.

Coefficient $l_1$

The coefficient $l_1$ above is called the Lyapunov coefficient

Expression of the coefficients

The coefficients $a,b$ above are computed as follows[Haragus]:

$$$a=\left\langle\mathbf{F}_{11}(\zeta)+2 \mathbf{F}_{20}\left(\zeta, \Psi_{001}\right), \zeta^{*}\right\rangle,$$$$$$l_1=\left\langle 2 \mathbf{F}_{20}\left(\zeta, \Psi_{110}\right)+2 \mathbf{F}_{20}\left(\bar{\zeta}, \Psi_{200}\right)+3 \mathbf{F}_{30}(\zeta, \zeta, \bar{\zeta}), \zeta^{*}\right\rangle.$$$

where

\begin{aligned} -\mathbf{J} \Psi_{001} &=\mathbf{F}_{01} \\ (2 i \omega-\mathbf{J}) \Psi_{200} &=\mathbf{F}_{20}(\zeta, \zeta) \\ -\mathbf{J} \Psi_{110} &=2 \mathbf{F}_{20}(\zeta, \bar{\zeta}). \end{aligned}

and where

$$$\mathbf{F}(x,p)-\mathbf{J}x := \sum_{1\leq q+l\leq p}\mathbf{F}_{ql}(x^{(q)},p^{(l)})+o(\|u\|+\|p\|)^p.$$$

with $\mathbf{F}_{ql}$ a $(q+l)$-linear map.

Normal form computation

The normal form (E) is automatically computed as follows

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

where prob is a bifurcation problem. 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 above call returns an object of type Hopf.

Note

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

Predictor

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

BifurcationKit.predictorMethod
predictor(hp, ds; verbose, ampfactor)


This function provides prediction for the periodic orbits branching off the Hopf bifurcation point.

Arguments

• bp::Hopf the bifurcation point
• ds at with distance relative to the bifurcation point do you want the prediction. Can be negative. Basically the new parameter is p = bp.p + ds.

Optional arguments

• verbose display information
• ampfactor = 1 factor multiplied to the amplitude of the periodic orbit.

Returned values

• t -> orbit(t) 2π periodic function guess for the bifurcated orbit.
• amp amplitude of the guess of the bifurcated periodic orbits.
• ω frequency of the periodic orbit (corrected with normal form coefficients)
• period of the periodic orbit (corrected with normal form coefficients)
• p new parameter value
• dsfactor factor which has been multiplied to abs(ds) in order to select the correct side of the bifurcation point where the bifurcated branch exists.
source

References

• Haragus

Haragus, Mariana, and Gérard Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer London, 2011. https://doi.org/10.1007/978-0-85729-112-7.