Cyclicity versus center problem. (English) Zbl 1210.34049
The authors consider a one-parameter family of differential systems of the form
\[
\begin{aligned} \dot x &=-y+ a^k x(x^2+ y^2)+ aP(x,y,a),\\ \dot y &=x+ a^ky(x^2+ y^2)+ aQ(x,y,a),\end{aligned}
\]
where \(P\) and \(Q\) are analytic functions, starting at least with terms of degree 4 in \(x\) and \(y\), and \(k\geq 1\) is an integer number.
It is proved that the cyclicity of the origin is at most \(k- 1\) and there are analytic functions, \(P\) and \(Q\), for which this upper bound is sharp.
It is proved that the cyclicity of the origin is at most \(k- 1\) and there are analytic functions, \(P\) and \(Q\), for which this upper bound is sharp.
Reviewer: A. P. Sadovskii (Minsk)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 37C27 | Periodic orbits of vector fields and flows |
Keywords:
polynomial differential system; focal quantities; Lyapunov constants; cyclicity; limit cyclesReferences:
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