Scaling Analysis at Transition of Chaos Driven by Euler's Numerical Algorithm

Jinde Cao, Ashish
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Abstract

Chaos is a nonlinear phenomenon that reveals itself everywhere in nature and in many fields of science. It has gained increasing attention from researchers and scientists over the last two decades. In this article, the nature of the fixed and periodic states are examined for a discrete two-parameter map; a composition of Euler’s numerical map and the logistic map. Further, the dynamical properties such as fixed states, period-doubling, and stability in fixed and periodic states are also described and the onset of chaos is characterized in detail followed by a few lemmas and remarks. Afterward, some scaling methods such as the bifurcation scale, fork-width scale, and Lyapunov exponent are illustrated to examine the appearance of chaos for the discrete two-parameter map. Experimental and numerical simulations are conducted followed by some bifurcation graphs, tables, and remarks. The scaling property is discussed in two key parameters. In addition, a comparative analysis of fork-width length, bifurcation length, and the maximum Lyapunov exponent is also presented to demonstrate the validity of the results.
由欧拉数值算法驱动的混沌转换的尺度分析
混沌是一种非线性现象,它在自然界和许多科学领域中无处不在。在过去的二十年里,它越来越受到研究人员和科学家的关注。本文研究了离散双参数映射的固定态和周期态的性质;欧拉数值映射和逻辑映射的组合。此外,还描述了系统的动力学特性,如固定状态、周期加倍、固定状态和周期状态下的稳定性,并详细描述了混沌的开始,随后给出了一些引理和注释。然后,举例说明了一些标度方法,如分岔标度、叉宽标度和Lyapunov指数,以检查离散双参数映射的混沌外观。进行了实验和数值模拟,并给出了一些分岔图、表和注释。讨论了两个关键参数的标度特性。此外,通过对分叉长度、分叉长度和最大Lyapunov指数的比较分析,验证了所得结果的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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