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Regularity properties of the period function near a center of a planar vector field. (English) Zbl 0769.34033
In case a planar vector field has a center, the annular region surrounding the center is filled with a one parameter family of periodic solutions. The function that assigns to a parameter value the period of the corresponding periodic solution is called the period function. The qualitative and analytic properties of this function play a role in the study of nonlinear boundary value problems and in bifurcation theory. The subject of the paper under review is the regularity properties of the period function at the center. Necessary and sufficient conditions are given, in terms of the blow up of the center, for the period function of a \(C^ \infty\) vector field to have a continuous extension to the center. For the analytic case when there is an essential singularity at the center, the growth properties of the period function at the center are explained in terms of some geometric properties. These results are applied to the problem of isochronicity of the period annulus. In particular, the results are used to answer a question posed by R. Conti [Processi di controllo lineari in \(\mathbb{R}^ n\) (1985; Zbl 0587.49001)]. He showed that a cubic system of the form \[ \dot x=y+P_ 2(x,y)+P_ 3(x,y), \dot y=Q_ 2(x,y)+Q_ 3(x,y) \] (the subscript denotes the degree of the indicated homogeneous form) where \(P_ 2^ 2+Q_ 2^ 2\neq 0\), \(P_ 3^ 2+Q_ 3^ 2\neq 0\) can have a center. He asked if such a center can be isochronous. This is shown to be impossible in the paper under review.

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