Datseris George, Wagemakers Alexandre
Max Planck Institute for Meteorology, 20146 Hamburg, Germany.
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles 28933, Madrid, Tulipán s/n, Spain.
Chaos. 2022 Feb;32(2):023104. doi: 10.1063/5.0076568.
We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the dynamical system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincaré maps, and projections of high-dimensional dynamics to a lower-dimensional space. The method is accompanied by a performant open-source implementation in the DynamicalSystems.jl library. The performance of the method outclasses the naïve approach of evolving initial conditions until convergence to an attractor, even when excluding the task of first identifying the attractors from the comparison. We showcase the power of our implementation on several scenarios, including interlaced chaotic attractors, high-dimensional state spaces, fractal basin boundaries, and interlaced attracting periodic orbits, among others. The output of our method can be straightforwardly used to calculate concepts, such as basin stability and final state sensitivity.
我们提出了一种全自动方法，该方法无需对动力学进行近似即可识别吸引子及其吸引域。该方法通过在动力系统流之上定义一个有限状态机来工作。该方法的输入是一个动力系统演化规则和一个划分状态空间的网格。不需要对吸引子的数量、位置或性质有先验知识。该方法适用于任意高维的动力系统，包括离散和连续的。它也适用于频闪映射、庞加莱映射以及高维动力学到低维空间的投影。该方法在DynamicalSystems.jl库中有一个高性能的开源实现。即使在比较中排除了首先识别吸引子的任务，该方法的性能也优于将初始条件演化到收敛到吸引子的简单方法。我们在几种场景中展示了我们实现的强大功能，包括交错混沌吸引子、高维状态空间、分形盆地边界和交错吸引周期轨道等。我们方法的输出可以直接用于计算诸如盆地稳定性和最终状态敏感性等概念。
