The zero-Hopf bifurcations of a four-dimensional hyperchaotic system

We consider the four-dimensional hyperchaotic system x?=a(y?x), y?=bx+u?y?xz, z?=xy?cz, and u?=?du?jx+exz, where a, b, c, d, j, e are real parameters. This extends famous Lorenz to four dimensions was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). characterize values of parameters for which their equilibrium points zero-Hopf points. Using averaging theory, we obtain sufficient conditions existence periodic orbits bifurcating from these equilibria give some examples illustrate conclusions. Moreover, stability given using Routh–Hurwitz criterion.