Two examples of stochastic webs that are generated by the web map (1.24) are shown in figure 1.3.

The figures each show the first iterates of the web map for a single initial value. In figure 1.3a, for (), the iterates form a regular web-like pattern, which in this case is called a
Certain
(approximate)
symmetry properties of the stochastic web
for
are obvious:
in this particular case the web
is characterized by translational symmetry in two transversal directions
(e.g. given by
) and
by rotational symmetry with respect to several classes of 2-fold
rotations, in addition to several reflection
and glide reflection
symmetries.
In fact it turns out that the underlying
*skeleton*
of the web is even invariant under
2-fold, 3-fold and 6-fold rotations
about
suitably chosen
centres of rotation.
See subsection 1.2.2 for
a rigorous definition of the skeletons of stochastic webs,
and for more information on the symmetry groups of these skeletons.

In figure 1.3b, for (), the situation is
different:
although this phase portrait
still reveals
approximate
rotational invariance with respect to the origin
(rotations through the angle ),
there is no
translational invariance
as
seen for example
in figure 1.3a.
In this sense,
the phase portrait in figure 1.3b is less
regular than that in figure 1.3a.
Webs in phase space with just rotational symmetry -- like the web for
-- are called *aperiodic* stochastic webs, as opposed to the
*periodic* stochastic webs
exhibiting
both
rotational *and* translational
symmetry, an example being the web for .

Having introduced periodic and aperiodic stochastic webs, the importance of the resonance condition (1.23) may now be illustrated by considering a value of with noninteger . Figure 1.4 shows a phase portrait of the POINCARÉ map (1.21) for .

Although this value of is fairly close to , the phase portraits for these two values of are significantly different (cf. figure 1.3a). In the nonresonance case the phase space structures reveal much less regularity: the whole phase plane -- with the exception of a region near the origin where some invariant lines persist -- forms a single dynamically connected chaotic region without an obvious inner structure, apart from variations in the density of points which are probably due to cantori.
In
the following
subsection 1.2.1 I present an argument for the fact
that periodic webs,
characterized by both rotational and translational symmetries,
can develop not for all
but only for a few
specific values of , namely
for

I then proceed in subsection 1.2.2 to the discussion of the skeleton, or ``backbone'', of the web and derive an equation that determines this overall structure of the web. Finally, in subsections 1.2.3 and 1.2.4 I briefly give an overview on what is known about the classical unbounded diffusive dynamics within the channels of the web that form around its skeleton; in particular I discuss the typical energy growth of a diffusing particle and the width of the channels of diffusive dynamics which is directly controlled by the kick strength . The results presented in subsections 1.2.2 and 1.2.3 are mainly based on [ZSUC91].

- Symmetries of the Web
- The Skeleton of the Web
- The Influence of the Kick Amplitude
- Diffusive Energy Growth in the Channels