  Geodesic motion has significant characteristics of space-time. We calculate
the principle Lyapunov exponent (LE), which is the inverse of the instability
timescale associated with this geodesics and Kolmogorov-Senai (KS) entropy for
our rotating Kerr-Kiselev (KK) black hole. We have investigate the existence of
stable/unstable equatorial circular orbits via LE and KS entropy for time-like
and null circular geodesics. We have shown that both LE and KS entropy can be
written in terms of the radial equation of innermost stable circular orbit
(ISCO) for time-like circular orbit. Also, we computed the equation marginally
bound circular orbit, which gives the radius (smallest real root) of marginally
bound circular orbit (MBCO). We found that the null circular geodesics has
larger angular frequency than time-like circular geodesics ($Q_o >
Q_{\sigma}$). Thus, null-circular geodesics provides the fastest way to
circulate KK black holes. Further, it is also to be noted that null circular
geodesics has shortest orbital period $(T_{photon}< T_{ISCO})$ among the all
possible circular geodesics. Even null circular geodesics traverses fastest
than any stable time-like circular geodesics other than the ISCO.
