Asymptotics in a family of linked strip maps

作者：Reeve Black Heather; Vivaldi Franco^*

来源：Physica D: Nonlinear Phenomena , 2015, 290: 57-71.

DOI：10.1016/j.physd.2014.09.003

• Round-off errors
• Pseudochaos
• Strip maps

We apply round-off to planar rotations, obtaining a one-parameter family of invertible maps of a two-dimensional lattice. As the angle of rotation approaches pi/2, the fourth iterate of the map produces piecewise-rectilinear motion, which develops along the sides of convex polygons.
We characterise the dynamics - which resembles outer billiards of polygons - as the concatenation of so-called strip maps, each providing an elementary perturbation of an underlying integrable system. Significantly, there are orbits which are subject to an arbitrarily large number of these perturbations during a single revolution, resulting in the appearance of a novel discrete-space version of near-integrable Hamiltonian dynamics.
We study the asymptotic regime of the limiting integrable system analytically, and numerically some features of its very rich near-integrable dynamics. We unveil a dichotomy: there is one regime in which the nonlinearity tends to zero, and a second where it does not. In the latter case, numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics; in the former case the fluctuations result in an intricate structure of resonances.

• 出版日期2015-1-1