The Skeleton of the Web

The iteration of the web map (1.24) for $q=4$ ($T=\pi /2$) typically yields phase portraits like those shown in figure 1.7.

Figure 1.7: Time evolution of the stochastic web for $q=4$ ($T=\pi /2$) and $V_0=1.0$. In figures (a) through (e) the initial value $(x_0,p_0)^t=(0,3.14)^t$ is iterated $10^3$/ $3\!\cdot\!10^3$/$10^4$/ $3\!\cdot\!10^4$/$10^5$ times, respectively. Figure (f) is a magnification of the square in the centre of figure (e); in addition, the first $10^3$ iterates of the points marked by $\blacktriangle$ are plotted.

In figures 1.7a through 1.7e one can observe the stochastic web for $M_4$ evolving under the iteration of a single initial value. Figure 1.7f shows how the irregular part of the phase portrait -- the channels that develop around the skeleton of the web -- and the regular part -- the meshes or cells of the web -- are intertwined with each other: each mesh of the web is densely filled^1.5with invariant lines, i.e. cross-sections of invariant tori of the underlying four-dimensional dynamics (in a generalized phase space spanned by $x$, $p$, $t$ and $H$). Only a few of these invariant lines are shown here. The dynamics in the channels is chaotic, whereas the invariant lines within the meshes indicate regular, nonchaotic behaviour. A situation with such a mixed phase portrait with coexisting regions of regular and chaotic dynamics is often characterized as soft or weak chaos [Gut90,SUZ88]. The origin of phase space is a fixed point of $M_4$; this is already clear from equation (1.29). Figure 1.7f indicates that all other elliptic points $(l\pi,m\pi)^t$, with $l+m$ even, are periodic with period four.

Figure 1.8, where phase portraits of the web map for several values of the kick amplitude $V_0$ are shown, allows the conclusion that the overall structure of the web is the same, regardless of the value of $V_0$ -- the main consequence of increasing the value of $V_0$ being a broadening of the channels of the web.

Figure 1.8: The stochastic webs for $q=4$ ($T=\pi /2$) and several values of the kick amplitude $V_0$. In figures (a) through (f) $V_0$ takes on the values $0.5/1.0/1.5/2.0/2.5/5.0$, respectively.

The stochastic webs for $q=4$ shown in figure 1.8 are (approximately) characterized by translational symmetry in two linearly independent directions (e.g. given by $(2\pi,0)^t$ and $(0,2\pi)^t$) 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 $T=2\pi/q$ with $q\in {\mathcal Q}$ -- 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 $x,p$ 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 $\vartheta,J$. The combination of both transformations can be described by the time-dependent generating function

which is of GOLDSTEIN's $F_2$-type [Gol80]. The old and new variables are related via

and the new Hamiltonian is obtained as

$$H(\vartheta,J,t) \; = \; V_0 \cos\left( \sqrt{2J}\sin(\vartheta+t) \right) \sum_{n=-\infty}^\infty \delta(t-nT).$$ (1.20)

Here, for simplicity, I use the same symbol $H$ for the Hamiltonian before and after the transformation. Since the new frame of reference is made to rotate with unit angular velocity -- which is just the angular velocity of the unperturbed oscillator after the scaling (1.15) -- only the kick contribution of the original Hamiltonian remains in (1.36).

Setting $n=:k+lq$ with $k\in\{1,2,\dots,q\}$, $l\in\mathbb{Z}$, and using the resonance condition (1.23) I have

$$H(\vartheta,J,t) \; = \; V_0 \sum_{k=1}^q \cos\left\{ \sqr... ...fty \delta \left( t-2l\pi-\frac{2k\pi}{q} \right), % \! , \,$$

because in this frame of reference all the summands contribute only at the times of kicks. Then, using the spectral decomposition of the OT2wncyrSh-function (cf. equation (1.2)),

$$\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\selectfont Sh}}(... ...; = \; \frac{1}{2\pi}\left(1+2\sum_{l=1}^\infty\cos lt\right),$$ (1.22)

the Hamiltonian is split into two components in a natural way:

Both ${\mathcal H}_q$ and ${\mathcal V}$ depend on both parameters $q$ and $V_0$. But in addition to separating the averaged Hamiltonian and the perturbation term, the splitting (1.39) also essentially separates the effects of the parameters: $q$ takes a decisive role for ${\mathcal H}_q$ where it alone determines the shape of the skeleton of the web, whereas changing $V_0$ does not influence the shape of the skeleton at all.^1.6For ${\mathcal V}$, on the other hand, $V_0$ is the more important parameter that controls the strength of the perturbation to the time averaged Hamiltonian. It turns out that it is mainly $V_0$ that determines the width of the diffusive channels that form around the skeleton of the web. I return to this issue in subsection 1.2.3, after having discussed the skeleton itself.

For the present purpose I need ${\mathcal H}_q$ expressed in terms of the original phase space variables $x,p$. Substitution of the transformation (1.35) into the Hamiltonian (1.39b) yields

$${\mathcal H}_q(x,p,t) \; = \; \frac{V_0}{2\pi} \sum_{k=1}^q ... ... \right) + p \sin \left( \frac{2k\pi}{q}-t \right) \right\},$$ (1.22)

where again I use the same symbol ${\mathcal H}_q$ for the Hamiltonian as a function both of $\vartheta,J$ and $x,p,t$. Since this expression is unpleasant in that it contains the time $t$ explicitly, I restrict the attention to just the times of kicks,

$$t_n \; := \; nT \; = \; \frac{2n\pi}{q} , \quad n\in\mathbb{Z}.$$ (1.23)

This is not a severe restriction because ${\mathcal H}_q$ is supposed to explain features of the web map, which is defined at kick times only. With the unit vectors

$$\vec{e}_{q,k} \; := \; \left( \begin{array}{@{}c@{}} \disp... ...e \sin\frac{2k\pi}{q} \end{array} \right), \quad k=1,\dots,q,$$ (1.24)

the averaged Hamiltonian now reads

$${\mathcal H}_q(x,p,t_n) \; = \; \frac{V_0}{2\pi} \sum_{k=1}... ...left\{ {x \choose p}\cdot % \hat{k} \vec{e}_{q,k} \right\},$$ (1.25)

which does not depend on $n$ any more; this means that the averaged dynamics given by ${\mathcal H}_q$ is the same for all times of kicks. Using equation (1.43), the skeletons of the stochastic webs can now be identified as level lines ${\mathcal H}_q(x,p,t_n)=\mbox{const.}(q)$, where the constant has to be chosen suitably for each $q$.

In subsection 1.2.3 I demonstrate that ${\mathcal V}$ can indeed be treated as a perturbation to ${\mathcal H}_q$. This means that for $V_0\to 0$ the dynamics of the web map is confined to the neighbourhood of surfaces -- i.e. lines in the $(x,p)$-plane -- of constant ${\mathcal H}_q(x,p,t_n)$. For each sufficiently small $V_0$ 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 ${\mathcal H}_q$-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 ${\mathcal H}_q$.

The case $q=4$ ($T=\pi /2$) may serve to exemplify the above statements. For this value of $q$ the time averaged Hamiltonian

$${\mathcal H}_4(x,p,t_n) \; = \; \frac{V_0}{\pi} \left( \cos x+\cos p \right)$$ (1.26)

is of the muffin-tin type and describes the classical analogue of the HARPER model [Har55,Ket92]; its level lines are plotted in figure 1.10b.

The ${\mathcal H}_4$-contours are dominated by the square grid defined by

$$p \; = \; \pm x + (2k+1)\pi, \quad k\in\mathbb{Z},$$ (1.27)

corresponding to ${\mathcal H}_4=0$. This square grid is the skeleton of the stochastic web for $q=4$.

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 $M_4$. Going backwards from figure 1.8f to 1.8a, $V_0$ 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 ${\mathcal H}_4$, 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 ${\mathcal H}_q$ works fine to explain the structure of the web's skeleton. Note that for increasing values of $V_0$ 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 $q=4$ and $q=3/6$, respectively (the skeleton (1.48) is given below when discussing the level lines of ${\mathcal H}_3$) -- to channels with increasingly wavy boundaries, as the approximation $H\approx{\mathcal H}_q$ 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 $q=4$ (figure 1.10b), the symmetry group is $p4m$ (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 $(l\pi,m\pi)^t$, $l+m$ even, in the centres of the meshes and the points of channel crossings: $(l\pi,m\pi)^t$, $l+m$ odd) and a class of 2-fold rotation symmetries (rotations about $((l+1/2)\pi,(m+1/2)\pi)^t$), in addition to the obvious translation and (glide) reflection symmetries.

Some short remarks about the other values of $q\in {\mathcal Q}$ might be in order. Equation (1.43) again illustrates the triviality of the cases $q=1$ and $q=2$ ($T=2\pi$ and $T=\pi$). For these $q$ there is no $p$-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, $q=3$ and $q=6$ ($T=2\pi /3$ and $T=\pi /3$) are also intertwined with each other,

and thus yield the same web skeleton, namely the level lines given by ${\mathcal H}_3(x,p,t_n)=-V_0/2\pi$:

see figure 1.10a. The symmetry group of this skeleton is $p6m$ 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 $q\not\in {\mathcal Q}$ show a more complicated structure. In figures 1.11a and 1.11b contours for a single ${\mathcal H}_q$-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 $V_0$, as can be observed, for example, in figure 1.8.

Footnotes

... 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 $V_0$-dependence of the web-generating Hamiltonian ${\mathcal H}_q$ 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 ${\mathcal H}_q$ being $V_0$-dependent, the geometry of the web generated by ${\mathcal H}_q$ does not depend on this parameter, since $V_0$ enters as a mere multiplicative factor into equation (1.39b).