A particle bouncing between a fixed and an oscillating
wall is an example of a dynamical system which shows
a mixture of regular and chaotic behavior.
One can study this dynamical system through a
Poincaré map
.
The variable u is the speed of the particle just
before hitting at time t the wall in position cos(t).
The image point is determined by the speed u_{1} of
the particle when it leavs the wall the next time
t_{1}. The time variable is taken modulo
so that P is an areapreserving map of the halfcylinder
.

The return map P is smooth on X if one assumes
that the moving wall imparts momentum (  sin(t))
to the ball but stays at a fixed position.
Let l be the average distance between the fixed and moving
wall and let v the speed of the particle. Assuming no gravity,
an approximation to P is
This simplified Fermi map is the composition of two maps,
the instantaneous velocity gain
at the impact time t and the following free motion
until the next impact.

The fixed points
of F are labeled
by a positive integer m. They correspond to initial speeds which lead
to a periodic return after m flooroscillations, hitting when the
floor is at a lowest rsp. highest point (the animation on this page
is such a fixed point with m=1). A linearization in the variable
v near
using
1/(v+u) = 1/v  1/v^{2} u + O(u ^{2}) leads to the map
with parameter
.
After a change of variables
,
this is
which is conjugated by the discrete involutive Legendre transformation
(x,y)=(q,pq) to the
Standard map
