Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type
The authors study the cyclicity of singular cycles (graphics) through a nilpotent singularity of saddle or elliptic type of codimension 3 inside \(C^{\infty}\) families of planar vector fields depending on parameters. A singular point \(p\) of a smooth planar vector field \(X\) is called a triple nilpotent point of saddle or elliptic type if in some local chart \((x,y):(\mathbb{R}^2,p)\to (\mathbb{R}^2,0)\) the vector field takes the form \[ X=y\frac{\partial}{\partial x}+(\varepsilon _1x^3+dx^4+bxy+ax^2y+ex^3y) \frac{\partial}{\partial y} +O(|(x,y)|^5), \] where, for the saddle case, \(\varepsilon _1=1\) holds and, for the elliptic case, \(\varepsilon _1=-1\) with \(b>2\sqrt{2}\). Let \(\Gamma\) be a graphic of \(X\) through \(p\). The cyclicity of \((X,p,\Gamma)\) is studied by considering a three-parameter unfolding \(X_{\mu}\) of \(X\). Cyclicity means the maximum number of limit cycles that may appear by perturbation inside the family \(X_{\mu}\). Finite cyclicity is closely related to the Hilbert-Arnold problem: Prove that, for any \(n\), the maximum cyclicity \(B(n)\) of nontrivial polycycles occurring in generic \(n\)-parameter families of planar vector fields is finite. Here, the finite cyclicity is proved for several of the most generic graphics through a nilpotent singularity of saddle or elliptic type of codimension 3. In some cases the results are independent of the exact codimension of the singular point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, all limit periodic sets are obtained. Different types of Dulac maps in the blown-up family are calculated, and a new general method to prove that some regular transition maps have a nonzero higher derivative at a point is developed. The finite cyclicity theorems are derived by a generalized derivation-division method introduced by R.~Roussari.
scientific article; zbMATH DE number 1763736
scientific article