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On the number of critical points of the period. (English) Zbl 0594.34028

It is shown that for the equation \(\ddot u+f(u)=0\), where \(f\) is a cubic polynomial, the period map can have at most three critical points. This is done by deriving the Picard-Fuchs equation for the period and analyzing the behavior of its solutions. One could make the conjecture that the maximal number is one. This would be the sharpest possible result without conditions on the coefficients of f. Applications are mentioned in the bifurcation theory of steady-state solutions.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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