Liénard limit cycles enclosing period annuli, or enclosed by period annuli. (English) Zbl 1084.34037
The author constructs polynomial systems of differential equations on \(\mathbb{R}^2\) of the form
\[
\text{(L)}\quad \dot x = y, \;\dot y = -g(x) -yf(x),
\]
of two types: (1) the system is of degree \(4k+2\) and has a period annulus (an annular region composed of periodic orbits) that surrounds \(2k\) limit cycles (isolated closed orbits); (2) the system is of degree \(6k\) and has \(k\) concentric limit cycles surrounding a center (a period annulus whose inner boundary is an equilibrium point). The author also proves existence of a real analytic system of the form (L) containing a center surrounded by infinitely many limit cycles.
Reviewer: Douglas S. Shafer (Charlotte)
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 |
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