Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems

In this paper, we perform a complete study of the Hopf bifurcations in the three-parameter Lorenz system, $$\dot{x} = \sigma (y-x),\,\dot{y} = \rho x - y - xz,\,\dot{z} = -bz + xy$$x˙=σ(y-x),y˙=ρx-y-xz,z˙=-bz+xy, with $$\sigma , \rho , b \in \mathbb {R}$$σ,ρ,b∈R. On the one hand, we reobtain the results found in the literature for the Lorenz model when the three parameters are positive. On the other hand, we completely determine the loci of all the degeneracies exhibited by the Hopf bifurcation of the origin and of the nontrivial equilibria. In this last case, we demonstrate, among other things, that the first two Lyapunov coefficients simultaneously vanish in two codimension-three bifurcation points, giving rise in both cases to the coexistence of three periodic orbits involved in a cusp bifurcation. The analytical study that we carry out, where several complicated expressions have to be handled, successfully closes the problem of the Hopf bifurcations in the Lorenz system. Moreover, from our results, it is easy to obtain all the information on the Hopf bifurcations in the Chen and Lü systems, taking into account that they are, generically, particular cases of the Lorenz system, as can be seen by means of a linear scaling in time and state variables.