  We aim at an understanding of the dynamical properties of a periodically
driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM)
at zero temperature, which is capable to show complex hysteresis. The system is
a combination of a continuous (harmonic oscillator) and a discrete (RFIM)
subsystem, which classifies it as a hybrid system. In this paper we focus on
the hybrid nature of the system and consider only independent spins in quenched
random local fields, which can already lead to complex dynamics such as chaos
and multistability. We study the dynamic behavior of this system by using the
theory of piecewise-smooth dynamical systems and discontinuity mappings.
Specifically, we present bifurcation diagrams, Lyapunov exponents as well as
results for the shape and the dimensions of the attractors and the
self-averaging behavior of the attractor dimensions and the magnetization.
Furthermore we investigate the dynamical behavior of the system for an
increasing number of spins and the transition to the thermodynamic limit, where
the system behaves like a driven harmonic oscillator with an additional
nonlinear smooth external force.
