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, ind_bif;    verbose = false, lens = getlens(br),    start_with_eigen = Val(true),    # Val(false): kernel basis via bordered systems    bls = MatrixBLS(), bls_adjoint = bls)

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 (nev, ζs, scaleζ, ...), we refer to get_normal_form. The result returns an object of type Cusp.

Returned object

The call get_normal_form(br, ind_bif) returns a Cusp point with the following fields

• x0, params, lens: the bifurcation point, the full parameter set and the two parameter axes,
• ζ (resp. ζ★): the real right (resp. left) vector spanning the kernel of $\mathbf L$ (resp. of $\mathbf L^{T}$), normalized so that $\langle \zeta,\zeta*\rangle = 1$,
• nf: a named tuple holding
  • nf.c: the coefficient $c$ of (E).

For completeness, the coefficient is computed as

\[c = \frac{1}{6}\left(\langle p, \mathbf C(q,q,q)\rangle + 3\langle p, \mathbf B(q, H_2)\rangle\right)\]

where $\mathbf B$, $\mathbf C$ are the second and third derivatives of $\mathbf F$ at $(x_0,p_0)$ and $H_2$ solves the bordered system $\mathbf L H_2 = \langle p, \mathbf B(q,q)\rangle\, q - \mathbf B(q,q)$ with $\langle p, H_2\rangle = 0$.

References