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1 Flows vs. Maps     continued...

On the other hand, dynamical systems are often modelled as maps, e.g., [2,3]

where we define the column vectors

The symbol N labels the iteration of the map. Comparing Eqs. (3), (4), (6), and (7), we see that the iteration index N can, in some cases, be considered a label for the time (taken in discrete ``jumps'').

Indeed, a map can be generated from a flow by taking:

where is the ``time delay'' [2]. In the right hand part of Eq. (8) we indicate the iterations corresponding to the discrete flow times in the left hand side. In other examples, maps can arise from flows by taking various types of Poincaré sections, in which one or more variables are fixed [2]. However, in many cases maps have been constructed as pure dynamical systems, without recourse to an underlying system of flows. Also, just as with flows, one can determine whether the map is conservative or dissipative [2,3].