Local Bifurcation and Center Problem for a More Generalized Lorenz System

In this paper, Hopf bifurcation and center problem are investigated for a class of more generalized Lorenz systems, which are Z_2 symmetric and quadratic three-dimensional systems. Firstly, the singular point quantities of one equilibrium are calculated carefully, and the two symmetric fourth-order weak foci are found. Secondly, the corresponding invariant algebraic surfaces are figured out, and the center conditions on a center manifold are determined. In this way it is proved that there exist at most eight small limit cycles from the two symmetric equilibria via a Hopf bifurcation, which is a new result for general Lorenz models. At the same time, when the center conditions are satisfied, the complete classification of Darboux invariants is established for this system.