  In this paper, we study the $T$-periodic solutions of the parameter-dependent
$\phi$-Laplacian equation \begin{equation*} (\phi(x'))'=F(\lambda,t,x,x').
\end{equation*} Based on the topological degree theory, we present some
atypical bifurcation results in the sense of Prodi--Ambrosetti, i.e.,
bifurcation of $T$-periodic solutions from $\lambda=0$. Finally, we propose
some applications to Li\'enard-type equations.
