Generalized Web Maps

The choice of for the kick function as in equation (1.7) is not the only one that leads to the emergence of stochastic webs. Similarly, there are other systems than the harmonic oscillator that generate stochastic webs under the influence of appropriate periodic impulsive driving forces. It is the purpose of the present subsection to give some background information about how the web map (1.24) can be generalized in such a way that it still produces stochastic webs, and to provide some according references.

In the present study I concentrate the attention on the prototype of stochastic webs, which is generated by the harmonic oscillator submitted to sinusoidal kicks. The classical dynamics of this model has been studied extensively by ZASLAVSKY and co-workers; their combined effort has culminated in the review [ZSUC88] and the monograph [ZSUC91]; see also the references therein. Some other contributions to this field are [LW89,Low91,Low92,SJM92], where mainly the diffusive properties of the system are studied.

In a natural generalizing step one may pass from sinusoidal kick functions to more general periodic ones, where the latter include sinusoidal kicks as a special case. This path has been taken for example in [YP92,Hov92,Lam93,Low96,DK96]. Among the kick functions studied are the square wave and the saw tooth functions, as well as other piecewise linear functions. In all these cases variations of stochastic webs can be found. In [DA95] the sine function is again chosen for the kick, but here with a shift in the argument: ; depending on the choice of , the stochastic web may or may not persist.

Even the condition of periodicity of the kick function can be dropped as in [Jun95], without necessarily destroying the web.

Another straightforward generalization of the web map concerns its
extrapolation into
phase spaces of higher dimensions (typically not more than 4).
See for example
[ZZN^{+}89,Zas91,ASZ91,Jun95,Hau97].
As their lower-dimensional counterparts, these higher dimensional maps are
most frequently studied with respect to their diffusive properties.

A more exotic discussion can be found in [LS87]. There, the relativistic analogue of the web map (1.24) is studied, with the result that the particle (in the underlying model system) can only be accelerated up to a critical energy. In other words, the corresponding web is finite. Still, the phase space structures close to the origin are the same as those generated by the map (1.24). The connection between the relativistic and nonrelativistic cases is further investigated in [HA99,AH00].

A limit beyond which no generalization can be pursued without spoiling the stochastic web is illustrated in [Vec95], where the effect of smoothing the -functions in the Hamiltonian (1.12) is investigated. The web is shown to be completely destroyed if the duration of the perturbation is arbitrarily short but finite, as opposed to being -shaped.

Until now I have only discussed the kicked
*harmonic oscillator*.^{1.3}Omitting
the harmonic oscillator potential, it is also possible to
consider a *free* particle under the influence of periodic kicks.
This approach is chosen, for example, in [SK91] and
[Sch93] for particles moving in one- and three-dimensional
position space, respectively.
Finally, rather than neglecting the oscillator potential in the
Hamiltonian (1.12) one can also *add* a term, for example
a centrifugal term that is obtained when discussing the radial
motion of a particle in three dimensions [Hip94].
In all these cases stochastic webs can be found.

Summarizing, it is quite obvious that the emergence of stochastic webs is a much more general phenomenon than it appears at first sight; the kicked harmonic oscillator with sinusoidal kicks as in equation (1.17) is thus found to be a typical representative of a much larger class of systems. Having made the above remarks about the conditions of existence of these webs, I now turn to a detailed description of these objects of the present study.

- ... oscillator.
^{1.3} - It may be mentioned in passing that even the well-known HÉNON maps -- in their area-preserving, dissipative and logistic limit variants [Hén76,Hén83] -- can be identified as POINCARÉ maps of a kicked harmonic oscillator with suitable kick functions [Hea92]. But in these particular cases no webs emerge.