Normal form of the Cusp bifurcation

We follow the paper[Kuznetsov] and consider a Cauchy problem

\[\dot x=\mathbf F(x,p).\]

We denote by $\mathbf L$ the jacobian of $\mathbf F$ at the bifurcation point $(x_0,p_0)$. We choose a basis such that:

\[\mathbf L q=0, \quad \mathbf L^{T} p=0, \quad \langle p, q\rangle=1.\]

Under some conditions, $x(t)\approx x_0+ w(t)q$ where $w$ satisfies the normal form:

\[\dot{w}=c w^{3}+O\left(w^{4}\right).\tag{E}\]

Normal form computation

The normal form (E) can be automatically computed as follows

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

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 Cusp.


You should not need to call get_normal_form except if you need the full information about the branch point.


  • Kuznetsov

    Kuznetsov, Yu. A. “Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE’s.” SIAM Journal on Numerical Analysis 36, no. 4 (January 1, 1999): 1104–24.