Universal centres and composition conditions

In this paper, we characterize the universal centres of the ordinary differential equations, where ai(θ) are trigonometric polynomials, in terms of the composition conditions. These centres are closely related with the classical Poincaré centre problem for planar analytic differential systems.Additionally, we show that the notion of universal centre is not invariant under changes of variables, and we also provide different families of universal centres. Finally, we characterize all the universal centres for the quadratic polynomial differential systems. © 2012 London Mathematical Society.