In this work, we investigate the presence of sub-diffusive behavior in the Chirikov–Taylor Standard Map. We show that trajectories started from special initial conditions, close to unstable periodic orbits, exhibit sub-diffusion due to stickiness, and can be modeled as a continuous-time random walk. Additionally, we choose a variant of the Ulam method to numerically approximate the Perron–Frobenius operator for the map, allowing us to calculate the exponent of anomalous diffusion by solving an eigenvalue problem and comparing its time dependence to the solution of the fractional diffusion equation. The results here corroborate other findings in the literature of anomalous transport in Hamiltonian maps and can be suitable to describe transport properties of other dynamical systems.

在这项工作中，我们调查了 Chirikov-Taylor 标准图中亚扩散行为的存在。我们表明轨迹从特殊的初始条件开始，接近不稳定的周期轨道，由于粘性而表现出亚扩散，并且可以建模为连续时间随机游走。此外，我们选择了 Ulam 方法的一个变体来对地图的 Perron-Frobenius 算子进行数值近似，使我们能够通过求解特征值问题并将其时间依赖性与分数扩散方程的解进行比较来计算异常扩散的指数。这里的结果证实了哈密顿地图中异常输运文献中的其他发现，并且可以适用于描述其他动力系统的输运特性。