  Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying
some decay condition and some locally log-H\"older continuity. In this article,
via first establishing characterizations of the variable exponent Hardy space
$H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the
Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain
its intrinsic square function characterizations including the intrinsic
Littlewood-Paley $g$-function, the intrinsic Lusin area function and the
intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure
characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the
variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in
terms of the intrinsic function is also presented.
