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 $p=p_0+\delta p$.
The coefficient $l_1$ above is called the Lyapunov coefficient
Expression of the coefficients
The coefficients $a,l_1$ 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
.
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.predictor
— Methodpredictor(hp, ds; verbose, ampfactor)
This function provides prediction for the periodic orbits branching off the Hopf bifurcation point.
Arguments
bp::Hopf
the bifurcation pointds
at with distance relative to the bifurcation point do you want the prediction. Can be negative. Basically the new parameter isp = bp.p + ds
.
Optional arguments
verbose
display informationampfactor = 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 valuedsfactor
factor which has been multiplied toabs(ds)
in order to select the correct side of the bifurcation point where the bifurcated branch exists.
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.