We obtain existence of $T$-periodic solutions to a second order system ofordinary differential equations of the form\[u^{\prime\prime}+cu^{\prime}+g(u)=p\]where $c\in\mathbb{R},$ $p\in C(\mathbb{R},\mathbb{R}^{N})$ is $T$-periodicand has mean value zero, and $g\in C(\mathbb{R}^{N},\mathbb{R}^{N})$ is e.g.sublinear. In contrast with a well known result by Nirenberg [6], whereit is assumed that the nonlinearity $g$ has non-zero uniform radial limits atinfinity, our main result allows rapid rotations in $g$.