  We explore the chaotic behavior of particle motion in a black hole with
quasi-topological electromagnetism. The chaos bound is found to be violated in
the higher order expansion of the metric function and the electric potential
near the horizon. We draw the Poincare sections of particle motion
corresponding to the chaos bound violated and non-violated cases, respectively.
Then we study the relationship between the maximal Lyapunov exponent \lambda_s
defined by the static equilibrium and the Lyapunov exponent of the particle
geodesic motion near the Reissner-Nordstrom(RN) black hole and the black hole
with quasi-topological electromagnetism. We find an interesting relationship
between the Lyapunov exponent \lambda_{ph} of photon's radial falling into the
black hole and the maximal Lyapunov exponent \lambda_s. For the black holes
whose metric function increases monotonically with radius outside horizon, this
leads to \lambda_{ph} \geq 2\lambda_s.
