  In this paper we are concerned with a homogeneous differential operator $p$
of order $m$ of which characteristic set of order $m$ is assumed to be a smooth
manifold. We define the Gevrey strong hyperbolicity index as the largest number
$s$ such that the Cauchy problem for $p+Q$ is well-posed in the Gevrey class of
order $s$ for any differential operator $Q$ of order less than $m$. We study
the case of the largest index and we discuss in which way the Gevrey strong
hyperbolicity index relates with behaviors of bicharacteristics of $p$ near the
characteristic manifold.
