  We analyze the motion of a {\it massless} and {\it chargeless} particle very
near to the event horizon. It reveals that the radial motion has exponential
growing nature which indicates that there is a possibility of inducing chaos in
the particle motion of an integrable system when it comes under the influence
of the horizon. This is being confirmed by investigating the Poincar$\acute{e}$
section of the trajectories with the introduction of a harmonic trap to confine
the particle's motion. Two situations are investigated: (a) {\it any} static,
spherically symmetric black hole and, (b) spacetime represents a stationary,
axisymmetric black hole (e.g., Kerr metric). In both cases, the largest
Lyapunov exponent has upper bound which is the surface gravity of the horizon.
We find that the inclusion of rotation in the spacetime introduces more chaotic
fluctuations in the system. The possible implications are finally discussed.
