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 DDE problem

\[\dot x = \mathbf{F}(x_t,p)\]

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 $(x,p)$.

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 = br.param_lens)

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 a point with information needed to compute the bifurcated branch.

mutable struct Hopf{Tv, T, Tω, Tevr, Tevl, Tnf} <: BifurcationPoint
	"Hopf point"
	x0::Tv

	"Parameter value at the Hopf point"
	p::T

	"Frequency of the Hopf point"
	ω::Tω

	"Right eigenvector"
	ζ::Tevr

	"Left eigenvector"
	ζstar::Tevl

	"Normal form coefficient (a = 0., b = 1 + 1im)"
	nf::Tnf

	"Type of Hopf bifurcation"
	type::Symbol
end

References