Dynamical Interpretation of Logistic Map using Euler’s Numerical Algorithm

Abstract

In the last two decades, the dynamics of difference and differential equations have found a celebrated place in science and engineering such as weather forecasting, secure communication, transportation problems, biology, the population of species, etc. In this article, we deal with the dynamical behavior of the logistic map using Euler’s numerical algorithm. The dynamical properties of Euler’s type logistic system are derived analytically as well as experimentally. In the analytical section, the dynamical properties such as fixed point, period-doubling, and irregularity are examined followed by s few theorems. Further, in the experimental section, the dynamical properties of Euler’s type logistic system are studied using period-doubling bifurcation plots. Because the dynamics of the Euler’s map depend on the Euler’s control parameter h, therefore, three major cases are discussed for all the dynamical properties for h = 0.1, 0.4, and 0.7. The result shows that as the value of parameter h decreases from 1 to 0 the growth rate parameter r increases rapidly. Therefore, the improved chaotic regime in bifurcation plots may improve the chaos based applications in science and engineering such as secure communication.

Keywords

• fixed point
• periodic orbit
• euler’s algorithm
• chaos