Awrejcewicz Jan, Krysko Anton V, Erofeev Nikolay P, Dobriyan Vitalyj, Barulina Marina A, Krysko Vadim A
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland.
Cybernetic Institute, National Research Tomsk Polytechnic University, 30 Lenin Avenue, 634050 Tomsk, Russia.
Entropy (Basel). 2018 Mar 6;20(3):175. doi: 10.3390/e20030175.
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations.
本文的目的是通过计算李雅普诺夫指数的不同方法（沃尔夫方法、罗森斯坦方法、坎茨方法、基于神经网络改进的方法以及同步方法），对由差分方程和微分方程所描述的经典问题（亨农映射、超混沌亨农映射、逻辑斯谛映射、罗斯勒吸引子、洛伦兹吸引子）进行分析，并同时使用傅里叶频谱和高斯小波。结果表明，对神经网络方法的改进使得计算李雅普诺夫指数谱成为可能，进而能够检测系统从规则动力学向混沌、超混沌及其他状态的转变。比较的目的是评估所考虑的算法，研究它们的收敛性，并确定针对特定系统类型和目标最合适的算法。此外，还提出了一种基于训练好的神经网络计算李雅普诺夫指数谱的算法。已证明所开发的方法对不同类型的系统都能产生良好的结果，并且不需要系统方程的先验知识。
