How are maps related to flows (differential equations)?
Every differential equation gives rise to a map, the time one map, defined by advancing the flow one unit of time. This map may or may not be useful. If the differential equation contains a term or terms periodic in time, then the time T map (where T is the period) is very useful–it is an example of a Poincaré section. The time T map in a system with periodic terms is also called a stroboscopic map, since we are effectively looking at the location in phase space with a stroboscope tuned to the period T. This map is useful because it permits us to dispense with time as a phase space coordinate: the remaining coordinates describe the state completely so long as we agree to consider the same instant within every period. In autonomous systems (no time-dependent terms in the equations), it may also be possible to define a Poincaré section and again reduce the phase space dimension by one. Here the Poincaré section is defined not by a fixed time interval, but by successive times when an orb
