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A codimension two bifurcation with a third order Picard-Fuchs equation. (English) Zbl 0571.34021

The authors show that for small \((\mu_ 1,\mu_ 2)\) lying in a certain cone the two parameter family of planar vectorfields \[ (1)\quad dx/dt=2xy+\mu_ 1x+\mu_ 2x^ 3,\quad dy/dt=1/4-x^ 2-y^ 2 \] has a unique limit cycle which is born from a Hopf bifurcation and dies in a saddle connection. Since (1) is a small perturbation of a Hamiltonian vectorfield \(X_ H\), applying the averaging method reduces the problem to showing that the function \[ \eta (h)=\int_{\gamma h}x^ 3 dy/\int_{\gamma h}x dy \] where \(\gamma_ h\) is a smooth compact component of the h level set of H, is strictly monotonic for \(-1/12<h<0\) or \(0<h<1/12\). This is done by showing that \(\eta\) satisfies a third order Picard-Fuchs equation.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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