Butusov Denis, Karimov Artur, Tutueva Aleksandra, Kaplun Dmitry, Nepomuceno Erivelton G
Youth Research Institute, Saint Petersburg Electrotechnical University "LETI", Saint Petersburg 197376, Russia.
Department of Computer Aided Design, Saint Petersburg Electrotechnical University "LETI", Saint Petersburg 197376, Russia.
Entropy (Basel). 2019 Apr 3;21(4):362. doi: 10.3390/e21040362.
In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Padé approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Padé approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.
在本文中，我们考虑基于微分方程解的直接帕德逼近的非线性积分技术及其在保守混沌初值问题中的应用。研究了通过非线性积分得到的离散映射的性质，包括相空间体积动力学、分岔图、谱熵和李雅普诺夫谱。我们还绘制了二维动力学图以阐明非线性积分技术所引入的特征。给出了经典积分方法和帕德逼近方法的对比研究。结果表明，非线性积分技术显著改变了非线性系统离散模型的行为，增加了李雅普诺夫指数和谱熵的值。这一特性降低了基于帕德逼近的数值方法在混沌系统模拟中的适用性，但它对于构建抗混沌退化的伪随机数生成器或具有高度非线性性质的离散映射仍然是有用的。
