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
with polar coordinates (which are identical to action-angle variables
of the unperturbed harmonic oscillator except for a sign in
while the second switches to a rotating
frame of reference with coordinates . The combination of both
transformations can be described by the
which is of GOLDSTEIN's -type [Gol80]. The old and new variables are related via
and the new Hamiltonian is obtained as
and using the resonance condition (1.23) I
For the present purpose I need
expressed in terms of the
original phase space variables . Substitution of the
transformation (1.35) into
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
the time averaged Hamiltonian
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.
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.