  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.
