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Quasiregularity property for unfoldings of hyperbolic polycycles

Quasiregularity property for unfoldings of hyperbolic polycycles

M. EL MORSALANI, A. MOURTADA, R. ROUSSARIE
  • Année : 1994
  • Tome : 222
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 58F21
  • Pages : 303-326
  • DOI : 10.24033/ast.263

Some years ago Yu. Ilyashenko proved that the return map of any planar analytic hyperbolic polycycle has a quasi-regularity property. This implies that the polycycle is isolated among limit cycles, a key step in the proof that any polynomial planar vector field has just a finite number of limit cycles. Here one proves a similar property for analytic one-parameter unfoldings of hyperbolic polycycles. As a consequence one deduces that some special unfoldings, with an unbrocken connection and a fixed product of again value ratios, have a finite cyclicity i.e., that the number of created limit cycles is bounded. Such unfoldings arrives for instance in quadratic vector fields, so that the result solves some of the cases in a general program formulated else where about the Hilbert's $16^{\mathrm {th}}$ Problem for quadratic vector fields.

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