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We consider the motion of test particles around a $Reissner-Nordstr\"{o}m$ black hole immersed into a strong external magnetic field modifying the spacetime structure. When the particles are neutral, their dynamics are nonintegrable because the magnetic field acts as a gravitational effect, which destroys the existence of a fourth motion constant in the $Reissner-Nordstr\"{o}m$ spacetime. A time-transformed explicit symplectic integrator is used to show that the motion of neutral particles can be chaotic under some circumstances. When test particles have electric charges, their motions are subject to an electromagnetic field surrounding the black hole as well as the gravitational forces from the black hole and the magnetic field. It is found that increasing both the magnetic field and the particle energy or decreasing the particle angular momentum can strengthen the degree of chaos regardless of whether the particles are neutral or charged. The effect of varying the black hole positive charge on the dynamical transition from order to chaos is associated with the electric charges of particles. The dynamical transition of neutral particles has no sensitive dependence on a change of the black hole charge. An increase of the black hole charge weakens the chaoticity of positive charged particles, whereas enhances the chaoticity of negative charged particles. With the magnitude of particle charge increasing, chaos always gets stronger., Comment: 16 pages, 11 figures

Based on the scalar-tensor-vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincare sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can do chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos., Comment: 12 pages,5 figures