Small-amplitude limit cycles of certain planar differential systems. (English) Zbl 1447.34031
Consider the planar system
\begin{align*}
\frac{dx}{dt}&= -y+xf(y), \\
\frac{dy}{dt}&= x+yf(x)\tag{1}
\end{align*}
under the assumption \(f(0)=0\) and that \(f\) is real analytic \(f(x)=\sum_{k=1}^{\infty}\alpha_kz^k.\)
The authors prove
- (i)
- The origin of (1) is a center if \(f\) is odd.
- (ii)
- Let \(f\) be of degree \(n\). Then the cyclicity of the origin is \(\big[\frac{n}{2}\big]\), that is, the maximum number of limit cycles which can bifurcate from the origin is \(\big[\frac{n}{2}\big]\), there is an analytic perturbation of (1) such that \(\big[\frac{n}{2}\big]\) limit cycles bifurcate from the origin.
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 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 |
| 34A34 | Nonlinear ordinary differential equations and systems |
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