There exist two essentially different approaches to the study of dynamical systems, based on
the following distinction:
time-continuous nonlinear differential equations ⇋ time-discrete maps
One approach starts from time-continuous differential equations and leads to time-discrete
maps, which are obtained from them by a suitable discretization of time. This path is
pursued, e.g., in the book by Strogatz [Str94]. 1 The other approach starts from the study of
time-discrete maps and then gradually builds up to time-continuous differential equations,
see, e.g., [Ott93, All97, Dev89, Has03, Rob95]. After a short motivation in terms of nonlinear
differential equations, for the rest of this course we shall follow the latter route to dynamical
systems theory. This allows a generally more simple way of introducing the important
concepts, which can usually be carried over to a more complex and physically realistic
context.
