Department of Engineering Science, The University of Electro-Communications, 1-5-1 Chofugaoka Chofu Tokyo 182-8585, Japan.
Department of Physics, Tokyo Metropolitan University, Minami-Osawa Hachioji, Tokyo 192-0397, Japan.
Phys Rev E. 2017 May;95(5-1):052207. doi: 10.1103/PhysRevE.95.052207. Epub 2017 May 11.
We have studied a two-dimensional piecewise linear map to examine how the hierarchical structure of stable regions affects the slow dynamics in Hamiltonian systems. In the phase space there are infinitely many stable regions, each of which is polygonal-shaped, and the rest is occupied by chaotic orbits. By using symbolic representation of stable regions, a procedure to compute the edges of the polygons is presented. The stable regions are hierarchically distributed in phase space and the edges of the stable regions show the marginal instability. The cumulative distribution of the recurrence time obeys a power law as ∼t^{-2}, the same as the one for the system with phase space, which is composed of a single stable region and chaotic components. By studying the symbol sequence of recurrence trajectories, we show that the hierarchical structure of stable regions has no significant effect on the power-law exponent and that only the marginal instability on the boundary of stable regions is responsible for determining the exponent. We also discuss the relevance of the hierarchical structure to those in more generic chaotic systems.
我们研究了一个二维分段线性映射，以研究稳定区域的层次结构如何影响哈密顿系统的慢动力学。在相空间中有无数的稳定区域，每个区域都是多边形形状，其余的区域被混沌轨道占据。通过使用稳定区域的符号表示，提出了一种计算多边形边缘的方法。稳定区域在相空间中呈层次分布，稳定区域的边缘显示出边缘不稳定性。再发性时间的累积分布符合幂律∼t^{-2}，与由单个稳定区域和混沌分量组成的系统相同。通过研究再发性轨迹的符号序列，我们表明稳定区域的层次结构对幂律指数没有显著影响，只有稳定区域边界上的边缘不稳定性负责确定指数。我们还讨论了层次结构与更通用的混沌系统中的层次结构的相关性。
