We have seen that the surfaces of section for a number of different
problems are qualitatively very similar. They all show two
qualitatively different types of motion: regular motion and
chaotic motion. They show that these types of orbits are clustered:
there are regions of the surface of section that have mostly
regular trajectories and other regions dominated by chaotic
behavior. We have also seen a transition to
large-scale chaotic behavior as some parameter is varied. Now we have
learned that the map that takes points on a two-dimensional surface of
section to new points on the surface of section is area preserving.
The sole property that these maps of the
section onto itself have in common (that we know of at this point) is
that they preserve area. Otherwise they are quite distinct. Suppose
we consider an abstract map of the section onto itself that is
area preserving, without regard for whether the map is generated by
some dynamical system. Do area-preserving maps show similar
phenomena, or is the dynamical origin of the map crucial to the
phenomena we have found?^{34}

Consider a map of the phase plane onto itself defined in terms of the dynamical
variables and its
``conjugate momentum'' *I*. The map is

This map is known as the ``standard map.''^{35}
A curious feature of the standard map is that the momentum variable
*I* is treated as an angular quantity. The
derivative of the map has determinant one, implying the map is area
preserving.

We can implement the standard map:

`(define ((standard-map K) theta I return failure)
(let ((nI (+ I (* K (sin theta)))))
(return ((principal-value :2pi) (+ theta nI))
((principal-value :2pi) nI))))
`

We use the `explore-map` procedure introduced earlier to use a
pointing device to interactively explore the surface of section. For
example, to explore the surface of section for parameter *K* = 0.6 we
use:

`(define window (frame 0.0 :2pi 0.0 :2pi))
(explore-map window (standard-map 0.6) 2000)
`

The resulting surface of section, for a variety of orbits chosen with the pointer, is shown in figure 3.27.

The surface of section does indeed look qualitatively similar to the surfaces of section generated by dynamical systems.

The surface of section for *K* = 1.4 (as shown in
figure 3.28) is dominated by a large
chaotic zone. The standard map exhibits a transition
to large-scale chaos near *K* = 1. So this abstract area-preserving map
of the phase plane onto itself shows behavior that is
similar to behavior in the sections generated by a Hamiltonian
dynamical system. Evidently, the area-preservation property of the
dynamics in the phase space plays a determining role for many
interesting properties of trajectories of mechanical systems.

**Exercise 3.13.** **Fun with Hénon's quadratic map**

Consider the map of the plane defined by the equations:

**a**. Show that the map preserves area.

**b**. Implement the map as a procedure. The interesting
range of *x* and *y* is ( `-` 1, 1). There will be orbits that escape.
You should check for values of *x* and *y* that escape from this range
and call the failure continuation when this occurs.

**c**. Explore the phase portrait of this map for a few values of
the parameter . The map is particularly interesting for
= 1.32 and = 1.2. What happens in between?

^{34} This question was also addressed in the remarkable paper
by Hénon and Heiles, but with a different map from what we use here.

^{35} The standard map has been extensively
studied. Early investigations were by
Chirikov [12] and by Taylor [44], so the
map is sometimes called the Chirikov-Taylor map. Chirikov coined the
term ``standard map,'' which we adopt.