<p>The subject of this paper concerns with the bifurcation of limit cycles for a four-dimensional hyperchaotic system. This hyperchaotic system depends on five parameters and we restrict our study to the set of parameters where a double zero-Hopf bifurcation may occurs, that is, where an isolated equilibrium point has a double zero and a pair of purely imaginary eigenvalues. For doing this study, some adequate changes of parameters and coordinates must be done in order that the computations become easier. The main result is based on the averaging theory.</p>