Bifurcations of cuspidal loops preserving nilpotent singularities

A cuspidal loop $L$ of a smooth planar vector field $X_0$ is a singular cycle formed by the union of a cuspidal singularity with 2-jet equivalent to $y\frac{\partial}{\partial x}+(x^2+b_0 xy)\frac{\partial}{\partial y}$ and a connection between its two local separatrices. We consider smooth unfoldings $X_\lambda$ along cuspidal loop $L$ of $X_0$ parameterized by $\lambda\in(\mathbb R^p,0)$. We assume that the cuspidal point exists at all parameter values. Let $P_0$ be the Poincaré map of $X_0$ along $L$. If this map is not formally equal to the identity, then it has the asymptotic expansion $P_0\colon u\to u+a_\pm|u|^\tau+\dotsb$, where $\pm$ is the sign of $u$, $a_\pm\neq 0$, and $\tau$ is a coefficient belonging to the sequence $S=\{1,7/6,11/6,2,\dots\}=\{n\in\mathbb N\}\cup\{m+1/6,\ m\in\mathbb N,\ m\ge 1\}\cup\{p-1/6,\ p\in\mathbb N,\ p\ge 2\}$. In this case we say that $(X_0,L)$ has finite codimension equal to the order of$\tau$ in the sequence $S$. The main result of this paper is that the cyclicity of the unfolding $X_\lambda$ has an explicit bound of $e.o._{\mathfrak H_0}(s)$ , where $s$ is the codimension of $(X_0,L)$ ($e.o._{\mathfrak H_0}(s)\sim\frac{5s}{3}$ when $s\to\infty$). This bound is sharp for generic unfoldings. For analytic unfoldings, the cyclicity is always finite and is given by the codimension of the related Abelian integral in the case of a non-conservative perturbation of a Hamiltonian vector field.