Lower bounds for the cyclicity of centers of quadratic three-dimensional systems

Abstract

We consider quadratic three-dimensional differential systems having a Hopf singular point. We study the cyclicity when the singular point is a center on the center manifold using higher order developments of the Lyapunov constants. As a result, we make a chart of the cyclicity by establishing the lower bounds for several known systems in the literature, among them the Rossler, Lorenz and Moon-Rand systems. Moreover, we obtain an example of a jerk system for which is possible to bifurcate 12 limit-cycles from the center, which is a new lower bound for three-dimensional quadratic systems.