We study the branching of the zero-set of $d(z,\la,\e)$ at multiple points when the parameter $\e$ varies. We apply the obtained results to improve the classical averaging theory for computing $T$-periodic solutions of $\lambda$-families of analytic $T$-periodic ordinary differential equations defined on $\mathbb{R}$, using the displacement functions $d(z,\la,\e)$ defined by these equations. We call the coefficients in the Taylor expansion of $d(z,\la,\e)$ in powers of $\e$ the averaged functions. The main contribution consists in analyzing the role that have the multiple zeros $z_0 \in \Omega$ of the first non-zero averaged function. The outcome is that these multiple zeros can be of two different classes depending on whether the zeros $(z_0, \lambda)$ belong or not to the analytic set defined by the real variety associated to the ideal generated by the averaged functions in the Noetheriang ring of all the real analytic functions at $(z_0, \lambda)$. We bound the maximum number of branches of isolated zeros that can bifurcate from each multiple zero $z_0$. Sometimes these bounds depend on the cardinalities of minimal bases of the former ideal. Several examples illustrate our results and they are compared with the classical theory, branching theory and also under the light of singularity theory of smooth maps. The examples range from polynomial vector fields to Abel differential equations and perturbed linear centers.