Finiteness for critical periods of planar analytic vector fields. (English) Zbl 0773.34032

The authors treat analytic vector fields on the plane and on the unit sphere which exhibit one-parameter families of periodic orbits. To any such “period annulus”, there corresponds a “period function” \(P(\xi)\), where \(\xi\in I\subset\mathbb{R}\) is a local parameter specifying the periodic orbits and \(P(\xi)\) denotes their period. \(\overline\xi\in I\) is said to be a critical period if \(P'(\overline\xi)=0\); and if \(P'(\xi)\equiv 0\), the period annulus is called isochronous. The main question investigated in this paper is to determine the cardinality of the set of critical periods. The basic result: If a relatively compact period annulus is not isochronous, then there are only finitely many critical periods. For the proof, previous work by the second author is used to desingularize the boundary of a period annulus (Blowing-up procedures transform a neighborhood of any boundary point to one of three canonical forms). The case of unbounded period annuli remains open in general. But for the special case of Hamiltonian vector fields with Hamiltonian \(H(x,y)={1\over 2}y^ 2+V(x)\), \(V\) a polynomial, the authors also prove the existence of at most finitely many critical periods unless \(V(x)\) is quadratic. The proof is by introducing appropriate hyperelliptic coordinates. The latter result is immediately applicable to the Neumann boundary value problem \(\ddot x+g(x)=0\), \(\dot x(0)=\dot x(T)=0\), where \(g(x)\) is a polynomial. It is proved that, unless \(g(x)\) is linear, this problem has at most a finite number of solutions whose number of zeros in \((0,T)\) is uniformly bounded. This result also holds for the polynomial Dirichlet problem \(\ddot x+g(x)=0\), \(x(0)=x(T)=a\); but its proof requires some modifications, since the solutions need no longer be periodic.
Reviewer: J.Hainzl (Kassel)


34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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