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

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.