A class of quadratic system with six limit cycles

In this paper, the Hopf bifurcation for a class of three-dimensional quadratic system is investigated by making use of singular values methods. Studied system lies in symmetrical vector field with regard to planar y = x and it has two symmetrical singular points (1,2,1) and (2,1,1). We give the expressions of the first three focal values of the singular point (1,2,1) and show that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can become a fine focus of third order at the same time. Moreover, we obtain that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can bifurcate three small limit cycles under a certain coefficient's disturbed condition. In sum, six limit cycles can bifurcate from system (1). In terms of Hopf bifurcation of a three-dimensional quadratic system, our results are new and interesting.