An important characteristic of a dynamical system is whether it is continuous or discrete. Continuous systems (often called flows) are given by differential equations whereas discrete dynamical systems (often called maps) are specified by difference equations [83]. Autonomous systems are characterized by the fact that input and output are omitted from the definition [72].
An important criterion for the analysis of a dynamical system is
whether it is time-dependent or not [41,42]. For
time-dependent dynamical systems the function that specifies
(continuous case) or
(discrete case) depends on the time itself whereas for
time-independent systems this function does not change over time.
For the analysis it is very important whether a dynamical system is linear or not. Linear dynamical systems are simple to analyze as opposed to non-linear systems, which typically do have intricate dynamical behavior [83]. Often linearization at specific locations is used to get insights into these complex non-linear dynamical systems.
Using linearization, another classification of dynamical systems
is crucial to separate simple cases from more complex ones.
Hyperbolic dynamical systems can be analyzed by linearization
efficiently, whereas non-hyperbolic systems may cause major
troubles in combination with
linearization [1,27]. Hyperbolic
systems are structurally stable, i.e., small perturbations of the
system parameters do not change the qualitative behavior of the
system. Non-hyperbolic systems are difficult to investigate,
occur rarely and can be considered the transitional phase between
two hyperbolic systems of different
nature [72].