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Liénard limit cycles enclosing period annuli, or enclosed by period annuli. (English) Zbl 1084.34037

The author constructs polynomial systems of differential equations on ${ℝ}^{2}$ of the form

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

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.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C07 Theory of limit cycles of polynomial and analytic vector fields
##### Keywords:
Liénard; limit cycle; period annulus