Centers and limit cycles for a family of Abel equations.

Given trigonometric monomials A 1 , A 2 , A 3 , A 4 , such that A 1 , A 3 have the same signs as sin ⁡ t , and A 2 , A 4 the same signs as cos ⁡ t , and natural numbers n , m > 1 , we study the family of Abel equations x ′ = ( a 1 A 1 ( t ) + a 2 A 2 ( t ) ) x m + ( a 3 A 3 ( t ) + a 4 A 4 ( t ) ) x n , a 1 , a 2 , a 3 , a 4 ∈ R . The center variety is the set of values a 1 , a 2 , a 3 , a 4 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 A 1 = A 3 and A 2 = A 4 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. [ABSTRACT FROM AUTHOR]