Centers and limit cycles for a family of Abel equations

Abstract

Given trigonometric monomials A1,A2,A3,A4, such that A1,A3 have the same signs as sin⁡t, and A2,A4 the same signs as cos⁡t, and natural numbers n,m>1, we study the family of Abel equations x′=(a1A1(t)+a2A2(t))xm+(a3A3(t)+a4A4(t))xn, a1,a2,a3,a4∈R. The center variety is the set of values a1,a2,a3,a4 such that the Abel equation has a center (every bounded solution is periodic). We prove that the codimension of the center variety is one or two. Moreover, it is one if and only if A1=A3 and A2=A4 and it is two if and only if the family has non-trivial limit cycles (different from x(t)≡0) for some values of the parameters.