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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, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf points. Using the averaging theory, we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria and give some examples to illustrate the conclusions. Moreover, the stability conditions of these periodic orbits are given using the Routh–Hurwitz criterion.