Liénard limit cycles enclosing period annuli, or enclosed by period annuli.

*(English)*Zbl 1084.34037The author constructs polynomial systems of differential equations on ${\mathbb{R}}^{2}$ of the form

$$\text{(L)}\phantom{\rule{1.em}{0ex}}\dot{x}=y,\phantom{\rule{4pt}{0ex}}\dot{y}=-g\left(x\right)-yf\left(x\right),$$

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)