Centers on center manifolds in the Lorenz, Chen and Lü systems

We provide in a very straightforward manner a proof for the existence of centers on center manifolds, for the generalized Lorenz system, x ̇ = a ( y - x ) , y ̇ = bx + cy - xz , z ̇ = dz + xy . From this result, the presence of this Hopf bifurcation of codimension infinity is trivially deduced for the Lorenz, Chen and Lü systems. Our outcomes are novel for the Lorenz and Chen systems and, in the case of the Lü system, we obtain again, but in an easier way, the result found in the literature. Moreover, we show for this Hopf bifurcation a relationship between the three systems.