Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane
The authors study the cyclicity for a 2-polycycle of the \(C^{\infty}\) planar system \[ \dot x=P_0(x,y)+\varepsilon P(x,y,\varepsilon ,\delta),\quad \dot x=Q_0(x,y)+\varepsilon Q(x,y,\varepsilon ,\delta), \tag{1} \] where \(\varepsilon >0\) is small and \(\delta\in U\subset \mathbb{R}^n\) with \(U\) bounded. For \(\varepsilon =0\) system (1) is supposed to have a 2-polycycle \(L=L_1\cup L_2\cup S_1\cup S_2\), where \(L_1\) and \(L_2\) are heteroclinic orbits connecting hyperbolic saddles \(S_1\) and \(S_2\). Assuming that for \(\varepsilon =0\) system (1) is integrable (has a \(C^1\) first integral), the authors find 1) conditions necessary for \(L\) to generate a limit cycle for some \(\delta =\delta_0\in U\) and \(\varepsilon\) small; 2) generic conditions under which system (1) has at most one or two limit cycles near \(L\) for \(\varepsilon +|\delta - \delta _0|\) small, i.e. conditions for \(L\) to have cyclicity 1 or 2.
scientific article; zbMATH DE number 1333006