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2.7] How are maps related to flows (differential equations)?


This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

2.7] 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 orbit crosses a fixed surface in
phase space. (Surface here means a manifold of dimension one less than the
phase space dimension).

However, not every flow has a global Poincaré section (e.g. any flow with an
equilibrium point), which would need to be transverse to every possible orbit.

Maps arising from stroboscopic sampling or Poincaré section of a flow are
necessarily invertible, because the flow has a unique solution through any
point in phase space--the solution is unique both forward and backward in
time. However, noninvertible maps can be relevant to differential equations:
Poincaré maps are sometimes very well approximated by noninvertible maps. For
example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the
logistic map, x -> -a+x^2.

It is often (though not always) possible to go backwards, from an invertible
map to a differential equation having the map as its Poincaré map. This is
called a suspension of the map. One can also do this procedure approximately
for maps that are close to the identity, giving a flow that approximates the
map to some order. This is extremely useful in bifurcation theory.

Note that any numerical solution procedure for a differential initial value
problem which uses discrete time steps in the approximation is effectively a
map. This is not a trivial observation; it helps explain for example why a
continuous-time system which should not exhibit chaos may have numerical
solutions which do--see [2.15].


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