Dynamics of a novel chaotic map

Abstract

In this study, another one-dimensional sinusoidal chaotic map around the bisector is constructed, which consists of two sine functions with irrational frequency ratios but comparable amplitude and phase. This design is motivated by the process equation. Investigating the unusual dynamics of this map and contrasting them with the process equation are the goals of this work. When two sine functions are included in a map, their interaction results in complex behaviors that have not been seen for the process equation. Like the process equation, this newly created map has an endless number of fixed points, but unlike the process equation, their stability cannot be fully defined. Furthermore, the suggested chaotic map can display a biotic-like time series when escaping regions are produced; however, unlike the process equation, these biotic-like time series only consist of transient components. In other words, the introduced map converges to a fixed point or periodic solution and never becomes unbounded because of the unlimited number of stable and unstable fixed points. The parameters of the map determine how long these biotic-like transient sections last. Additionally, coexisting attractors and multi-stability are seen in this map, much like in the process equation. The significant dynamics of this map are examined with time series, cobweb, bifurcation, and Lyapunov exponent diagrams. Tests are conducted before and after the creation of the escaping regions. Additionally, two-dimensional bifurcation diagrams are used to study the simultaneous impact of many pairs of factors on the map’s dynamics.