Periodic orbits in complex Abel equation

Abstract

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt = z 3 +B(t)z 2 +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 dz/dt = A(t)z 3 + B(t)z 2 studied in the literature, where the center variety is located in a finite number of connected components.