  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 $\{f(x,y)=0\} \cap \bar{B_R((0,0))}$ as a limit
periodic set. Furthermore, we show that if $f(x,y)$ is polynomial, then there
exists a polynomial family with these properties.

Dynamical Systems;Classical Analysis and ODEs