Sakellariou Konstantinos, Stemler Thomas, Small Michael
Complex Systems Group, Department of Mathematics & Statistics, The University of Western Australia, Crawley WA 6009, Australia.
Nodes & Links Ltd, Leof. Athalassas 176, Strovolos, Nicosia 2025, Cyprus.
Phys Rev E. 2021 Feb;103(2-1):022214. doi: 10.1103/PhysRevE.103.022214.
We propose a computationally simple and efficient network-based method for approximating topological entropy of low-dimensional chaotic systems. This approach relies on the notion of an ordinal partition. The proposed methodology is compared to the three existing techniques based on counting ordinal patterns-all of which derive from collecting statistics about the symbolic itinerary-namely (i) the gradient of the logarithm of the number of observed patterns as a function of the pattern length, (ii) direct application of the definition of topological permutation entropy, and (iii) the outgrowth ratio of patterns of increasing length. In contrast to these alternatives, our method involves the construction of a sequence of complex networks that constitute stochastic approximations of the underlying dynamics on an increasingly finer partition. An ordinal partition network can be computed using any scalar observable generated by multidimensional ergodic systems, provided the measurement function comprises a monotonic transformation if nonlinear. Numerical experiments on an ensemble of systems demonstrate that the logarithm of the spectral radius of the connectivity matrix produces significantly more accurate approximations than existing alternatives-despite practical constraints dictating the selection of low finite values for the pattern length.
我们提出了一种基于网络的计算简单且高效的方法，用于逼近低维混沌系统的拓扑熵。这种方法依赖于序数划分的概念。将所提出的方法与基于计数序数模式的三种现有技术进行了比较，所有这些技术都源于收集关于符号行程的统计信息，即：（i）观察到的模式数量的对数作为模式长度的函数的梯度；（ii）拓扑置换熵定义的直接应用；（iii）长度增加的模式的增长比。与这些方法不同，我们的方法涉及构建一系列复杂网络，这些网络在越来越精细的划分上构成基础动力学的随机近似。只要测量函数在非线性时包含单调变换，就可以使用多维遍历系统生成的任何标量可观测量来计算序数划分网络。对一组系统的数值实验表明，尽管实际限制要求为模式长度选择低有限值，但连通性矩阵的谱半径的对数产生的近似值比现有方法更准确得多。
