Simple bifurcation branch point
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
.
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.predictor
— Methodpredictor(bp, ds; verbose, ampfactor)
This function provides prediction for the zeros of the Transcritical bifurcation point.
Arguments
bp::Transcritical
the bifurcation pointds
distance to the bifurcation point for the prediction. Can be negative. Basically the parameter isp = bp.p + ds
Optional arguments
verbose
display informationampfactor = 1
factor multiplying prediction
Returned values
x0
trivial solution (which bifurcates)x1
non trivial guess, corrected with Lyapunov-Schmidt expansionp
new parameter valueamp
non trivial zero of the normal form (not corrected)xm1
non trivial guess for the parameterpm1
pm1
parameter valuebp.p - ds
BifurcationKit.predictor
— Methodpredictor(bp, ds; verbose, ampfactor)
This function provides prediction for the zeros of the Pitchfork bifurcation point.
Arguments
bp::Pitchfork
the bifurcation pointds
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 isbp.p + abs(ds) * dsfactor
wheredsfactor = ±1
depending on the criticality.
Optional arguments
verbose
display informationampfactor = 1
factor multiplying prediction
Returned values
x0
trivial solution (which bifurcates)x1
non trivial guessp
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.amp
non trivial zero of the normal form