  We study the homoclinic orbit and the violation of chaos bound, which are
obtained by particle motions around a black hole that coexist with anisotropic
matter fields. The homoclinic one is associated with an unstable local maximum
of the effective potential. By perturbing a particle located slightly away from
the homoclinic one, we numerically compute Lyapunov exponents indicating the
sensitivity of the initial value. Our results demonstrate that the violation of
the chaos bound increases with higher angular momentum, and the anisotropic
matter gives rise to violating the chaos bound further, even in the case of the
nonextremal black hole. We utilize the Hamiltonian-Jacobi formalism to
explicitly illustrate how the geodesic motion of a particle can be integrable
in the procedure of obtaining our findings.
