  A simple discontinuous map is proposed as a generic model for nonlinear
dynamical systems. The orbit of the map admits exact solutions for wide regions
in parameter space and the method employed (digit manipulation) allows the
mathematical design of useful signals, such as regular or aperiodic
oscillations with specific waveforms, the construction of complex attractors
with nontrivial properties as well as the coexistence of different basins of
attraction in phase space with different qualitative properties. A detailed
analysis of the dynamical behavior of the map suggests how the latter can be
used in the modeling of complex nonlinear dynamics including, e.g., aperiodic
nonchaotic attractors and the hierarchical deposition of grains of different
sizes on a surface.
