Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation d z / d t = z3 + B (t) z2 + C (t) z, where B and C are smooth, 2π-periodic complex valuated functions, t ∈ R and z ∈ C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations d z / d t = A (t) z3 + B (t) z2 studied in the literature, where the center variety is located in a finite number of connected components. © 2006 Elsevier Inc. All rights reserved.