  In this paper, the authors establish the boundedness for a large class of
intrinsic square functions $\mathcal{G}_{\alpha}$, $g_{\alpha}$,
$g^{\ast}_{\tilde{\lambda},\alpha}$ and their commutators
$[b,\mathcal{G}_{\alpha}]$, $[b,g_{\alpha}]$ and
$[b,g^{\ast}_{\tilde{\lambda},\alpha}]$ generated with $\lambda$-central $BMO$
functions $b\in CBMO^{p,\lambda}(\mathbb{R}^{n})$ on generalized central Morrey
spaces $\mathcal{B}^{q,\varphi}(\mathbb{R}^{n})$ for
$1<q<\infty,0<\alpha\leq1$, respectively. All of the results are new even on
the central Morrey spaces $\mathcal{B}^{q,\lambda}(\mathbb{R}^{n})$.
