  In this paper, we prove that for every index perfect non-degenerate compact
star-shaped hypersurface $\Sigma\subset{\bf R}^{2n}$, there exist at least $n$
non-hyperbolic closed characteristics with even Maslov-type indices on $\Sigma$
when $n$ is even. When $n$ is odd, there exist at least $n$ closed
characteristics with odd Maslov-type indices on $\Sigma$ and at least $(n-1)$
of them are non-hyperbolic. Here we call a compact star-shaped hypersurface
$\Sigma\subset {\bf R}^{2n}$ {\rm index perfect} if it carries only finitely
many geometrically distinct prime closed characteristics, and every prime
closed characteristic $(\tau,y)$ on $\Sigma$ possesses positive mean index and
whose Maslov-type index $i(y, m)$ of its $m$-th iterate satisfies $i(y, m)\not=
-1$ when $n$ is even, and $i(y, m)\not\in \{-2,-1,0\}$ when $n$ is odd for all
$m\in {\bf N}$.
