Stochastic Webs in

Classical Mechanics

Oh what a tangled web we weave ... |

Marmion |
---|

SIR WALTER SCOTT |

A *stochastic web*
is a web-like structure
in the phase space of a
classical
dynamical system
(normally with a single degree of freedom)
--
in the present case: of the kicked harmonic oscillator. This
structure completely covers the phase space and possesses certain
(approximate)
symmetry
properties: rotational and, in some cases, translational invariance. The
dynamics on the web can be given, for example, by ZASLAVSKY's
*web map*.
There are two distinct types of dynamics
in the web:
regular motion in the meshes of
the web and irregular, stochastic motion in the channels;
the second
provides the motivation for characterizing
the web as ``stochastic''.

Stochastic webs can be obtained by considering suitable discrete mappings derived from the classical dynamics of the kicked harmonic oscillator. I therefore begin the exposition of the subject in section 1.1 by introducing and briefly discussing this model system. Then, in section 1.2, I make use of the kicked harmonic oscillator to generate the stochastic webs that are in the focus of interest here and provide the classical background for the entire present study. Finally, in section 1.3, I address the complementary case characterized by nonexistence of stochastic webs.

- The Kicked Harmonic Oscillator
- Newtonian Equations of Motion
- Canonical Formulation
- Discrete Dynamics -- The Web Map
- Generalized Web Maps

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

- The Complementary Case: Nonresonance