DYNAMICAL INTERPRETATIONS OF A GENERALIZED CUBIC SYSTEM

Abstract

Chaotic map is a typical route in mathematics to describe the dynamical interpretations in various applications of science and engineering. However, the dynamics in the traditional logistic chaotic map $r\vartheta(1-\vartheta)$ depends on the single control parameter $r$. In this article, a generalized cubic chaotic map with three changeable parameters $a$, $b$ and $r$ is introduced and its dynamical properties are studied. The added new control parameter increases the flexibility in the system due to which it can fit in various applications. A few cases are discussed showing the effectiveness of the changeable parameters in various properties such as stationary and periodic states, stability in stationary states, Lyapunov exponent property, bifurcation interpretation, and the minimum entropy control. Further, the developments are illustrated mathematically as well as experimentally followed by period-doubling bifurcation and Lyapunov exponent diagrams. Moreover, it is noticed that as compared to traditional chaotic systems, a bifurcation self-similarity is seen along the x-axis in all the cases of the cubic map. Moreover, a brief summary on Shannon minimum entropy is also given to control the unstable stationary and periodic states in chaotic regime.