Hopf bifurcations in an extended Lorenz system

In this work, we study Hopf bifurcations in the extended Lorenz system, \(\dot{x}=y\), \(\dot{y}=mx-ny-mxz-px^{3}\), \(\dot{z}=-az+bx^{2}\), with five parameters \(m,n,p,a,b\in \mathbb{R}\). For some values of the parameters, this system can be transformed to the classical Lorenz system. In this paper, we give conditions for occurrence of Hopf bifurcation at the equilibrium points. We find totally three limit cycles, each of them located around one of the three equilibria of the system. Numerical simulations illustrate the validity of these conditions and the existence of limit cycles.