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 ($\delta p \equiv p-p_0$):

\[a_{01}\delta p + \frac{a_{02}}{2}\delta p^2 + z\left(b_{11}\delta p + \frac{b_{20}}{2}z + \frac{b_{30}}{6}z^2\right) = 0 \tag{E}\]

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 $a_{01}\neq 0$, this is a Saddle-Node bifurcation
if $a{01}=0,b_{20}\neq 0$, the bifurcation point is Transcritical and the bifurcated branch exists on each side of $p_0$.
if $a_{01}=0,b_{20}=0, b_{30}\neq 0$, the bifurcation point is a Pitchfork and the bifurcated branch only exists on one side of $p_0$.

Reduced equation 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.

Note

Strictly speaking, this is not a normal form but a reduced equation. However, to keep the API simple, we use the same name especially for get_normal_form because it also returns the Hopf/BT/... normal forms which are true normal forms.

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.predictor — Method

predictor(
    bp::Union{BifurcationKit.Transcritical, BifurcationKit.TranscriticalMap},
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:x0, :x1, :xm1, :p, :pm1, :dsfactor, :amp, :p0), <:NTuple{8, Any}}

Compute predictions for solution branches near a Transcritical bifurcation point.

This function predicts points on both the trivial and bifurcated branches near a Transcritical bifurcation based on the normal form coefficients.

Arguments

bp::Transcritical: Transcritical bifurcation point.
ds: Parameter distance from the bifurcation point. Can be positive or negative. The new parameter value will be p = bp.p + ds.

Keyword Arguments

verbose = false: Display prediction information.
ampfactor = 1: Multiplicative factor for the amplitude prediction.

Returns

A named tuple with the following fields:

x0: predicted point on the trivial solution branch
x1: predicted point on the bifurcated branch (forward direction)
xm1: predicted point on the bifurcated branch (backward direction)
p: new parameter value bp.p + ds
pm1: backward parameter value bp.p - ds
amp: amplitude of the bifurcated solution (from normal form, uncorrected)
p0: original bifurcation parameter value

Details

The predictor solves the normal form equation b1 * ds + b2 * amp / 2 = 0 to determine the amplitude of the bifurcated branch. The function handles two cases depending on whether the tangent vector aligns with the critical eigenvector.

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BifurcationKit.predictor — Method

predictor(
    bp::Union{BifurcationKit.Pitchfork, BifurcationKit.PitchforkMap},
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:x0, :x1, :p, :dsfactor, :amp, :δp), <:NTuple{6, Any}}

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

Arguments

bp::Pitchfork bifurcation point.
ds at with distance relative to the bifurcation point do you want the prediction. Based on the criticality of the Pitchfork, 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

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