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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 λ 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.

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Correspondence to André Belotto.

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Belotto, A. Analytic Varieties as Limit Periodic Sets. Qual. Theory Dyn. Syst. 11, 449–465 (2012). https://doi.org/10.1007/s12346-012-0070-4

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Keywords

  • Periodic Orbit
  • Implicit Function Theorem
  • Analytic Family
  • Connected Part
  • Planar Vector