One dimensional maps can help to understand ordinary differential equations::
If you cut the Lorentz attractor at a suitable place with a plane, the intersection
points form a pointset lies approximately on an interval. This means that if we look
at a solution curve which starts on a point of that interval, the next time this
curve will intersect the plane will again be on that interval. This return map is
essentially a map on the interval. Indeed, when changing the parameters of the differential
equation one finds bifurcation diagrams for periodic flow lines similar to the
bifurcation diagram for the logistic map.
