Intermittent maps of the interval are simple and widely-studied models for chaos with slow mixing rates, but have been notoriously resistant to numerical study. In this paper we present an effective framework to compute many ergodic properties of these systems, in particular invariant measures and mean return times. The framework combines three ingredients that each harness the smooth structure of these systems' induced maps: Abel functions to compute the action of the induced maps, Euler-Maclaurin summation to compute the pointwise action of their transfer operators, and Chebyshev Galerkin discretisations to compute the spectral data of the transfer operators. The combination of these techniques allows one to obtain exponential convergence of estimates for polynomially growing computational outlay, independent of the order of the map's neutral fixed point. This enables numerical exploration of intermittent dynamics in all parameter regimes, including in the infinite ergodic regime.
翻译:间隔期间的断断图是简单和广泛研究的混乱模式,混合率缓慢,但一直受到臭名昭著的阻力,无法进行数字研究。在本文中,我们提出了一个有效框架,用以计算这些系统的许多电子特性,特别是变异测量和平均返回时间。这个框架结合了三个要素,每个要素都利用这些系统引出地图的平稳结构:Abel函数,用来计算引出地图的动作;Euler-Maclaurin参数,用来计算其传输操作员的点性动作;Chebyshev Galerkin 离散,用来计算传输操作员的光谱数据。这些技术的结合使得一个人能够获得多球形增长计算外缘的指数性汇总估计数,这与地图的中性固定点的顺序无关。这使得可以对所有参数系统中的间歇动态进行数字探索,包括在无限的ergodic 系统中。