Scientists that are interested in dynamical systems (and the local analysis of these systems) are confronted with a lot of terms, formulas, and definitions. Non-mathematicians get easily confused by studying some of the relevant literature in the beginning. Differing terms for the same object do not help to clear up the situation as well as subtle differences in the interpretation of mathematical symbols do not simplify the understanding. This was one of the reasons to compile relevant terms that occur often in literature and to assemble the different definitions. For example, the curvature of a 3D curve can either be calculated from the Frenét formulas or by analyzing the Jacobian matrix of the dynamical system. On the other hand it is interesting to see how some (local) attributes of a dynamical system can be derived by rather different approaches. This seems to be especially useful when some of the straightforward techniques are not possible due to incomplete or insufficient specifications. One example is the analysis of dynamical systems that are given as sampled data which do not allow the use of straightforward analytical approaches in most cases.
Before terms and definitions that are relevant for the local
analysis of dynamical systems are discussed, some
high-level classifications of dynamical systems are listed.
Thereafter an arc from differential geometry aspects when
analyzing trajectories of dynamical systems is spanned to the
analysis of linear dynamical systems and its interpretation. In
this sections we present well known concepts but concentrate on
giving a unifying view of various terms and definitions, which are
sometimes used ambiguously and interchangeably in literature.
Then we discuss dynamical system
analysis near special subsets of the topology of behavior to end
up with a new approach to locally analyze points on trajectories.