Simple bifurcation branch point

A simple branch point $(x_0,p_0)$ for the problem $F(x,p)=0$ satisfies $\dim \ker dF(x_0,p_0) = 1$. At such point, we can apply Lyapunov-Schmidt reduction to transform the initial problem in large dimensions to a scalar polynomial ($\delta p \equiv p-p_0$):

\[a\delta p + z\left(b_1\delta p + \frac{b_2}{2}z + \frac{b_3}{6}z^2\right) = 0 \tag{E}\]

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

More precisely, if $\ker dF(x_0,p_0) = \mathbb R\zeta$, one can show that $x_0+z\zeta$ is close to a solution on a new branch, thus satisfying $F(x_0+z\zeta,p_0+\delta p)\approx 0$.

In the above scalar equation,

• if $a\neq 0$, this is a Saddle-Node bifurcation
• if $a=0,b_2\neq 0$, the bifurcation point is a Transcritical one where the bifurcated branch exists on each side of $p$.
• if $a=0,b_2=0, b_3\neq 0$, the bifurcation point is a Pitchfork one where the bifurcated branch only exists on one side of $p$. If it exists at smaller values then $p$, this is a subcritical Pitchfork bifurcation. In the other case, this is a supercritical Pitchfork bifurcation.

Normal form computation

The reduced equation (E) can be automatically computed as follows

getNormalForm(br::ContResult, ind_bif::Int ;
	verbose = false, ζs = nothing, lens = br.param_lens)

where prob is the 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 getNormalForm. The result returns the following:

mutable struct SimpleBranchPoint{Tv, T, Tevl, Tevr, Tnf} <: BranchPoint
	"bifurcation point"
	x0::Tv

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

	"Right eigenvector(s)"
	ζ::Tevr

	"Left eigenvector(s)"
	ζstar::Tevl

	"Normal form coefficients"
	nf::Tnf

	"Type of bifurcation point"
	type::Symbol
end