In Part I of this two-part paper we utilize stability-preserving mappi
ngs to identify general dynamical systems having equivalent qualitativ
e properties. We establish sufficient conditions for a mapping to have
stability-preserving properties and we employ stability-preserving ma
ppings to formulate a comparison theorem for general dynamical systems
. Qualitative aspects of dynamical systems which we address include st
ability, asymptotic stability, and exponential stability of invariant
sets (with an emphasis on equilibria) and boundedness of motions. The
results developed herein include most of the corresponding comparison
results reported in the literature as special cases. In addition, the
present results are applicable to certain classes of general dynamical
systems (such as discrete event systems) which cannot be addressed by
corresponding previous results. For such contemporary systems, the mo
tions which make up a dynamical system are not determined by equations
, as is required by the existing results. We demonstrate the applicabi
lity of the results developed herein by means of a class of discrete e
vent systems in Part II of this paper.