About a conjecture for uniformly isochronous polynomial centers
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
Computing center conditions for vector fields with constant angular speed
We investigate the planar analytic systems which have a center-focus equilibrium at the origin and whose angular speed is constant. The conditions for the origin to be a center (in fact, an isochronous center) are obtained. Concretely, we find conditions for the existence of a Cw-commutator of the field. We cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families (-y + x(H1 + Hm), x + y(H1 + Hm)t and (-y + x(H2 + H2n), x + y(H2 + H2n))t, with Hi homogeneous polynomial in x,y of degree i. In these cases, the maximum number of limit cycles which can bifurcate from a fine focus is determined.
Periodic solutions of a quartic nonautonomous equation
On considere des equations de la forme: z=z 4 +α(t)z 3 +β(t)z 2 +γ(t)z. On donne un ensemble qui a au moins huit solutions periodiques. Les coefficients sont des polynomes quadratiques en cost et sint