Non-simple branch point

We expose our method to study non-simple branch points. Such branch point $(x_0,p_0)$ for the problem $F(x,p)=0$ satisfies $d=\dim \ker dF(x_0,p_0) > 1$ and the eigenvalues have zero imaginary part. At such point, we can apply Lyapunov-Schmidt reduction to transform the initial problem in large dimensions to a $d$-dimensional polynomial equation, called the reduced equation.

More precisely, it is possible to write $x = u + v$ where $u\in \ker dF(x_0,p_0)$ and $v\approx 0$ belongs to a vector space complement of $\ker dF(x_0,p_0)$. It can be shown that $u$ solves $\Phi(u,\delta p)=0$ with $\Phi(u,\delta p) = (I-\Pi)F(u+\psi(u,\delta p),p_0+\delta p)$ where $\psi$ is known implicitly and $\Pi$ is the spectral projector on $\ker dF(x_0,p_0)$. Fortunately, one can compute the Taylor expansion of $\Phi$ up to order 3. Computing the bifurcation diagram of this $d$-dimensional multivariate polynomials can be done using brute force methods.

Once the zeros of $\Phi$ have been located, we can use them as initial guess for continuation but for the original $F$ !!

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 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. It returns a point with all requested information:

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

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

	"Right eigenvectors"
	ζ::Tevr

	"Left eigenvectors"
	ζstar::Tevl

	"Normal form coefficients"
	nf::Tnf

	"Type of bifurcation point"
	type::Symbol
end

Using the reduced equation

Once a branch point has been computed bp = get_normal_form(...), you can do all sort of things.

• For example, quoted from the file test/testNF.jl, you can print the 2d reduced equation as follows:

julia> BifurcationKit.nf(bp2d)
2-element Array{String,1}:
 " + (3.23 + 0.0im) * x1 * p + (-0.123 + 0.0im) * x1^3 + (-0.234 + 0.0im) * x1 * x2^2"
 " + (-0.456 + 0.0im) * x1^2 * x2 + (3.23 + 0.0im) * x2 * p + (-0.123 + 0.0im) * x2^3"

• You can evaluate the reduced equation as bp2d(Val(:reducedForm), rand(2), 0.2). This can be used to find all the zeros of the reduced equation by sampling on a grid or using a general solver like Roots.jl.

• Finally, given a $d$-dimensional vector $x$ and a parameter $\delta p$, you can have access to an initial guess $u$ (see above) by calling bp2d(rand(2), 0.1)

Predictor

The predictor for a non trivial guess at distance $\delta p$ from the bifurcation point is provided by the method