Chaos in the motion of a test scalar particle coupling to the Einstein tensor in Schwarzschild–Melvin black hole spacetime

We present firstly the equation of motion for a test scalar particle coupling to the Einstein tensor in the Schwarzschild–Melvin black hole spacetime through the short-wave approximation. Through analyzing Poincaré sections, the power spectrum, the fast Lyapunov exponent indicator and the bifurcation diagram, we investigate the effects of the coupling parameter on the chaotic behavior of the particles. With the increase of the coupling strength, we find that the motion of the coupled particle for the chosen parameters becomes more regular and order for the negative couple constant. While, for the positive one, the motion of the coupled particles first undergoes a series of transitions betweens chaotic motion and regular motion and then falls into horizon or escapes to spatial infinity. Our results show that the coupling brings about richer effects for the motion of the particles