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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 2 of the form

(L)x ˙=y,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.

34C05Location of integral curves, singular points, limit cycles (ODE)
34C07Theory of limit cycles of polynomial and analytic vector fields