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On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. (English) Zbl 0628.34032
Let \(X_{\lambda}\) be a family of vector fields on the plane such that \(X_ 0\) has a separatrix loop \(\Gamma\). This means that \(X_ 0\) has a hyperbolic saddle point \(s_ 0\) and that one of the stable separatrices of \(s_ 0\) coincides with one of its unstable ones. Suppose that div \(X_ 0(s_ 0)=0\) and that \(P_ 0(x)-x\) is not flat at \(x=0\), where \(P_ 0\) is the Poincaré map of \(X_ 0\) around \(\Gamma\) and x is a parameter on a transversal segment to \(\Gamma\), with \(x=0\) on \(\Gamma\). Then, for \(\lambda\) small enough, \(X_{\lambda}\) has a uniform finite number of limit cycles near \(\Gamma\). More precisely, if \(P_ 0(x)-x\) is equivalent to \(\beta_ kx^ k\) with \(k\geq 1\) and \(\beta_ k\neq 0\), this number is less than 2k; if \(P_ 0(x)-x\) is equivalent to \(\alpha_{k+1}x^{k+1}Log x\) with \(k\geq 1\) and \(\alpha_{k+1}\neq 0\), this number is less than \(2k+1\).

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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