The Skeleton of the Web

The iteration of the web map (1.24) for () typically yields phase portraits like those shown in figure 1.7.

Figure 1.8, where phase portraits of the web map for several values of the kick amplitude are shown, allows the conclusion that the overall structure of the web is the same, regardless of the value of -- the main consequence of increasing the value of being a broadening of the channels of the web.

The stochastic webs for shown in figure 1.8 are
(approximately)
characterized by translational symmetry in two linearly
independent directions (e.g. given by and ) and
by two rotational symmetries with respect to
2-fold and 4-fold rotation axes,
in addition to several (glide) reflection symmetries.
More can be said about the symmetries of the *skeleton* of this web
-- see below.

I now proceed to the explanation of these characteristics of the web dynamics that -- for the resonant values with -- manifest themselves in figures 1.3a, 1.7 and 1.8.

The overall structure of the web, its *skeleton*, can be explained by
splitting the Hamiltonian (1.17) into two parts, such that the first
gives an analytical description of the skeleton and the second
can be treated as a perturbation to the first.

In a preliminary step the Hamiltonian (1.17) is submitted to a pair
of successive canonical transformations, the first replacing the original
coordinates
with polar coordinates (which are identical to action-angle variables
of the unperturbed harmonic oscillator except for a sign in
the angle),
while the second switches to a rotating
frame of reference with coordinates . The combination of both
transformations can be described by the
time-dependent
generating function

which is of GOLDSTEIN's -type [Gol80].
The old and new variables are related via

and the new Hamiltonian is obtained as

Setting with
,
,
and using the resonance condition (1.23) I
have

(1.21) |

(1.22) |

Both and depend on both parameters and . But in addition to separating the averaged Hamiltonian and the perturbation term, the splitting (1.39) also essentially separates the effects of the parameters: takes a decisive role for where it alone determines the shape of the skeleton of the web, whereas changing does not influence the shape of the skeleton at all.

For the present purpose I need
expressed in terms of the
original phase space variables . Substitution of the
transformation (1.35) into
the Hamiltonian
(1.39b) yields

(1.22) |

(1.23) |

(1.24) |

which does not depend on any more; this means that the averaged dynamics given by is the same for all times of kicks. Using equation (1.43), the

In subsection 1.2.3 I demonstrate that can indeed be treated as a perturbation to . This means that for the dynamics of the web map is confined to the neighbourhood of surfaces -- i.e. lines in the -plane -- of constant . For each sufficiently small the skeleton of the web is then given by some specific contour lines of the time averaged Hamiltonian (1.43). The term ``specific'' is important here: the -level has to be chosen in such a way that the corresponding contour lines form an infinitely extended structure, in fact the skeleton of the stochastic web. The other levels are important, too; they approximate the bounded quasiperiodic motion in the meshes of the web and accordingly have the topological structure of circles in the phase plane. Figures 1.9-1.11 show contour plots for some of the more important .

The case () may serve to exemplify the above statements. For this
value of
the time averaged Hamiltonian

corresponding to . This square grid is the skeleton of the stochastic web for .

Figure 1.10b is to be compared with the phase portraits in figures 1.7 and 1.8 that have been generated by iteration of . Going backwards from figure 1.8f to 1.8a, tends to zero, and accordingly the stochastic webs more and more approach the rectangular grid of figure 1.10b. The quasiperiodic regular dynamics in the meshes of the web, depicted in figure 1.7f, is also (more or less) well approximated by the contour lines of , as can be seen in figure 1.12, where magnifications of figures 1.7f and 1.10b have been superimposed.

Summarizing, the time averaged Hamiltonian works fine to explain the structure of the web's skeleton. Note that for increasing values of one observes not only a broadening of the channels of the web, but also a transition from the straight channels -- described by the web-skeletons (1.45, 1.48) for and , respectively (the skeleton (1.48) is given below when discussing the level lines of ) -- to channels with increasingly wavy boundaries, as the approximation becomes worse.
In contrast to the inevitably
qualitative description of the
symmetry of the stochastic webs displayed in figures 1.3a and
1.7/1.8,
the symmetry groups of the
*skeletons*
of the webs can be determined
exactly, because with equation (1.43) a closed formula for
the pattern they are forming is available.
For the skeleton of stochastic webs with (figure 1.10b),
the symmetry group is
(using the ``international notation'' for planar space groups
[GS87]).
It is characterized by two classes of 4-fold rotation symmetries
(here: the centres of rotation being the elliptic points
, even, in the centres of the meshes and the points
of channel crossings: , odd)
and a class of 2-fold rotation symmetries
(rotations about
), in addition to the obvious
translation and (glide) reflection symmetries.

Some short remarks about the other values of
might be in order.
Equation (1.43) again illustrates the triviality of the
cases and ( and ).
For these there is no -dependence any more and

such that the phase space structures effectively become one-dimensional
rather than two-dimensional,
as already discussed in the previous subsection
and displayed in figure 1.9.

Similarly, and ( and ) are also
intertwined with each other,

and thus yield the same web skeleton, namely the level lines given by
:

see figure 1.10a.
The symmetry group of this skeleton is
and includes three classes of rotational symmetries, namely a
2-fold, a 3-fold and a 6-fold rotational symmetry (the rotations are
about the vertices and the centres of the triangles, and about the centres
of the hexagons in figure 1.10a, respectively),
in addition to the obvious translation and (glide) reflection symmetries.

The skeletons for show a more complicated structure. In figures 1.11a and 1.11b contours for a single -level each are shown; these levels are chosen in such a way that the contour lines cover as many hyperbolic points in the stochastic layer as possible, thereby coming fairly close to the underlying separatrix (cf. [ZSUC88]). Here, as in figure 1.3b, the aperiodic organization of phase space is clearly visible.

Essentially, the method I have used here to identify the webs' skeletons is an averaging procedure (see the derivation of the splitting (1.39)). LOWENSTEIN [Low92] shows that this can be viewed as just the first step of an iterative scheme which is similar to the BIRKHOFF-GUSTAVSON normalization method [Gus66,Eng93,ESE95] and, step by step, yields higher order approximations for the Hamiltonian (1.17). Using this scheme it is possible to systematically derive improved expressions describing the separatrices of the webs and thereby explain their waviness that develops for larger values of , as can be observed, for example, in figure 1.8.

- ... filled
^{1.5} -
This statement is not to be taken too literally.
In general, the interior of each mesh appears to be filled by just
invariant lines only when
studied
on a coarse scale. A more thorough investigation typically
exhibits
narrow chains of islands and the corresponding stochastic
layers,
especially in the vicinity of the stochastic regions of the
stochastic web itself
[CSU
^{+}87,LL92]. The stochastic channels, on the other hand, are in general also interspersed with islands (some of them arbitrarily small) of regular dynamics. - ...
all.
^{1.6} - It is interesting, though,
that there exists a
-dependence of the web-generating Hamiltonian
at all,
regardless of the scaling that is used.
This, a
consequence of the transformation (1.35) reducing the
Hamiltonian to its kick contribution,
indicates that the web-like structure in phase space is
the
*combined*result of both the electric and magnetic components of the field (1.1). In other words, the structure stems from the*interaction*of the corresponding translational and rotational symmetries. Despite being -dependent, the*geometry*of the web generated by does not depend on this parameter, since enters as a mere multiplicative factor into equation (1.39b).