  This paper is devoted to the study of infinitesimal limit cycles that can
bifurcate from zero-Hopf equilibria of differential systems based on the
averaging method. We develop an efficient symbolic program using Maple for
computing the averaged functions of any order for continuous differential
systems in arbitrary dimension. The program allows us to systematically analyze
zero-Hopf bifurcations of polynomial differential systems using symbolic
computation methods. We show that for the first-order averaging,
$\ell\in\{0,1,\ldots,2^{n-3}\}$ limit cycles can bifurcate from the zero-Hopf
equilibrium for the general class of perturbed differential systems and up to
the second-order averaging, the maximum number of limit cycles can be
determined by computing the mixed volume of a polynomial system obtained from
the averaged functions. A number of examples are presented to demonstrate the
effectiveness of the proposed algorithmic approach.
