John Lowenstein:
Pseudochaotic Webs of Kicked Oscillator Maps
Abstract:
Suppose that a one-dimensional harmonic oscillator is given instantaneous kicks 4 times per natural
period, with a kick amplitude which varies sinusoidally with position. Such a kicked oscillator, viewed
stroboscopically in phase space, has a Poincare section with 2-dimensional crystalline symmetry, and
chaotic orbits which extend to infinity within a so-called stochastic web. Chaos within the web has both
local and global manifestations. The Poincare map, folded into the fundamental cell of the 2D crystal,
has orbits within the web with positive Lyapunov exponent; the same web orbits, viewed globally on the
plane, move to infinity in a manner which resembles a random walk, so that the average behavior of a
statistical ensemble of orbits resembles diffusion.
The focus of my talk is recent work with G. Poggiaspalla, and F. Vivaldi on a version of the kicked
oscillator model in which the kick amplitude is a piecewise linear (sawtooth) function. This produces a
vanishing Lyapunov exponent, while leaving open the possibility of generating self-similar structures
for suitably chosen parameter values. Within such "pseudochaotic webs", orbits can move to infinity
according to asymptotic long-time power laws. For models with quadratic irrational parameter, we have
found the full range of possible behaviours: sub-diffusive, diffusive, super-diffusive and ballistic.
For a more complicated model with cubic irrational parameter, we have found a variety of power-law
behaviours within different ergodic components of the same pseudochaotic web.