Screen chaotic motion by Shannon entropy in curved spacetimes

We find a novel characteristic for chaotic motion by introducing Shannon entropy for periodic orbits, quasiperiodic orbits, and chaotic orbits. We compare our approach with the previous methods including Poincaré section, Lyapunov exponent, fast Lyapunov indicator, recurrence plots (Rps), and fast Fourier transform (FFT) for orbits around black holes immersed in magnetic fields, and show that they agree with each other quite well. The approach of Shannon entropy is intuitively clear, and theoretically reasonable since it becomes larger and larger from a periodic orbit to chaotic orbit. We demonstrate that Shannon entropy can be a powerful probe to distinguish between chaotic and regular orbits in different spacetimes, and reversely may lead to a new route to define the entropy for a single orbit in phase space, and to find more fundamental relations between thermodynamics and dynamics. Furthermore, we find that the fluctuations of entropy of chaotic orbits are stronger than those of order orbits.