  We present firstly the equation of motion for the test scalar particle
coupling to the Chern-Simons invariant in Kerr black hole spacetime by the
short-wave approximation. We have analyzed the dynamical behaviors of the test
coupled particles by applying techniques including Poincar\'e sections, fast
Lyapunov exponent indicator, bifurcation diagram and basins of attraction. It
is shown that there exists chaotic phenomenon in the motion of scalar particle
interacted with the Chern-Simons invariant in a Kerr black hole spacetime. With
the increase of the coupling strength, the motion of the coupled particles for
the chosen parameters first undergoes a series of transitions betweens chaotic
motion and regular motion and then falls into horizon or escapes to spatial
infinity. Thus, the coupling between scalar particle and Chern-Simons invariant
yields the richer dynamical behavior of scalar particle in a Kerr black hole
spacetime.
