Analysis of Dynamical Behavior of One Cubic Polynomial via Topological Conjugacy

ABSTRACT Dynamical systems have become a significant research area in recent years due to their rich structure and wide‐ranging applications. Analyzing their behavior often remains challenging, as even simple polynomial maps, such as the logistic map, can display highly complex dynamics. To circumvent this challenge, piecewise linear maps that are topologically conjugate to such polynomials are frequently considered. This article focuses on analyzing the dynamical behavior of a specific cubic polynomial. For this purpose, we first define a so‐called double‐sided tent map on a closed interval and represent it in base 2 to formulate its periodic points and investigate its dynamical properties. Furthermore, we construct a topological conjugacy map between the double‐sided tent map and the cubic polynomial, thereby demonstrating that the cubic polynomial also exhibits the same chaotic behavior as double‐sided tent map. We additionally prove that these dynamical systems possess periodic points of all even periods, but no periodic points of odd period except for a fixed point.