Majoration du nombre de zéros des solutions de certaines équations différentielles et applications géométriques
Résumé
Sturm theory permits to minorate for different types of closed curves the number of certain special points or of certain lines, as the number of vertices or of normals through a point. In this paper, we apply hedgehog theory to obtain results that permit to compare such numbers. The main result gives a majoration of the number of zeros of solutions of certain forced harmonic oscillator equations. As a geometric application, we compare for instance the number of vertices and the maximal number of normals through a point for $N$-hedgehogs of class $C_{+}^{3}$ in $\QTO{mathbb}{\mathbb{R}}^{3}$
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