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Bifurcations of cuspidal loops preserving nilpotent singularities
D. Panazzoloa,
R. Roussarieb a Universidade de São Paulo
b Université de Bourgogne
Abstract:
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.
Key words and phrases:
Bifurcations, cuspidal loop, nilpotent singularity.
MSC: Primary
34C23; Secondary
34C25,
34C37,
37F75 Received: April 4, 2003
Language: English
DOI:
10.17323/1609-4514-2005-5-1-207-244
Fulltext:
References