Analytic varieties as limit periodic sets

Abstract

Let $f(x,y)\not\equiv 0$ be a real-analytic planar function. We show that, for almost every $R>0$ there exists an analytic 1-parameter family of vector fields $X_{\lambda }$ which has $\{f(x,y)=0\}\cap \overline{B_R((0,0))}$ as a limit periodic set. Furthermore, we show that if $f(x,y)$ is polynomial, then there exists a polynomial family with these properties.

Cited by in corpus (3)

• Infinitesimal Poincaré-Bendixson Problem in dimension three
• Topological classification of limit periodic sets of polynomial planar vector fields
• Examples of infinitesimal non-trivial accumulation of secants in dimension three