Analytic varieties as limit periodic sets

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 $X_λ$ 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.

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