## 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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### References:

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