  We have investigated the motion of timelike particles along geodesic in the
background of accelerating and rotating black hole spacetime. We confirmed that
the chaos exists in the geodesic motion of the particles by Poincar\'e
sections, the power spectrum, the fast Lyapunov exponent indicator and the
bifurcation diagram. Moreover, we probe the effects of the acceleration and
rotation parameters on the chaotic behavior of a timelike geodesic particle in
the black hole spacetime. Our results show that the acceleration brings richer
physics for the geodesic motion of particles.
