# Simple bifurcation branch point

References

The general method is exposed in Golubitsky, Martin, David G Schaeffer, and Ian Stewart. Singularities and Groups in Bifurcation Theory. New York: Springer-Verlag, 1985, VI.1.d page 295

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 Transcritical and the bifurcated branch exists on each side of $p_0$.
• if $a=0,b_2=0, b_3\neq 0$, the bifurcation point is a Pitchfork and the bifurcated branch only exists on one side of $p_0$.

## Normal form computation

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

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

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 get_normal_form. The result returns an object of type BranchPoint.

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 methods (depending on the type of the bifurcation point)

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


This function provides prediction for the zeros of the Transcritical bifurcation point.

Arguments

• bp::Transcritical the bifurcation point
• ds distance to the bifurcation point for the prediction. Can be negative. Basically the parameter is p = bp.p + ds

Optional arguments

• verbose display information
• ampfactor = 1 factor multiplying prediction

Returned values

• x0 trivial solution (which bifurcates)
• x1 non trivial guess, corrected with Lyapunov-Schmidt expansion
• p new parameter value
• amp non trivial zero of the normal form (not corrected)
• xm1 non trivial guess for the parameter pm1
• pm1 parameter value bp.p - ds
source
BifurcationKit.predictorMethod
predictor(bp, ds; verbose, ampfactor)


This function provides prediction for the zeros of the Pitchfork bifurcation point.

Arguments

• bp::Pitchfork the bifurcation point
• ds at with distance relative to the bifurcation point do you want the prediction. Based on the criticality of the Picthfork, its sign is enforced no matter what you pass. Basically the parameter is bp.p + abs(ds) * dsfactor where dsfactor = ±1 depending on the criticality.

Optional arguments

• verbose display information
• ampfactor = 1 factor multiplying prediction

Returned values

• x0 trivial solution (which bifurcates)
• x1 non trivial guess
• 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.
• amp non trivial zero of the normal form
source