A novel chaotic map with a shifting parameter and stair-like bifurcation diagram: dynamical analysis and multistability

Abstract

In this paper, the behavior of a 1D chaotic map is proposed which includes two sine terms and shows unique dynamics. By varying the bifurcation parameter, the map has a shift, and the system's dynamics are generated around the cross points of the map and the identity line. The irrational frequency of the sine term makes the system have stable fixed points in some parameter intervals by increasing the bifurcation parameter. So, the bifurcation diagram of the system shows that the trend of the system's dynamics changes in a stair shape with slope one by changing the bifurcation parameter. Due to the achieving multiple steady states in some intervals of the parameter, the proposed system is known as multistable. The multistability dynamics of the map are investigated with the help of cobweb diagrams which reveal an interesting asymmetry in repeating parts of the bifurcation diagram.

Applying exponential unit for breaking symmetry of memristive maps

“…Discrete maps and chaotic maps are fundamental concepts in the realm of dynamical systems and nonlinear dynamics, with applications spanning various scientific disciplines and engineering fields [23][24][25][26]. Chaotic maps, in particular, are characterized by their sensitivity to initial conditions, leading to seemingly random and unpredictable behavior that conceals underlying patterns [27,28].…”

Dynamic Behavior and Bifurcation Analysis of a Modified Reduced Lorenz Model

“…Experimental results demonstrated superior security against statistical attacks and differential attacks compared to the classical encryption algorithm and the improved encryption algorithm. Furthermore, Mliki et al [7] introduced new chaotic maps with applications in stochastic processes, while Ramadoss et al [8] recently investigated the behaviors of a one-dimensional chaotic map comprising two sine terms. Overall, these references highlight the wide-ranging applications and significant contributions made in the field of chaotic mappings, underscoring their relevance and advancements in various areas of research.…”

A new chaotic map derived from the Hermite–Kronecker–Brioschi characterization of the Bring-Jerrard quintic form

“…There are actually more and more new chaotic maps that we proposed in the last decade (Alpar 2015, Alpar 2017, Alpar 2018a, ) also a novel chaotic scheme (Alpar 2018b) for one-predator one-pray Lotka-Volterra system. Besides our papers, there are a few papers proposing new chaotic maps with the necessary dynamical analysis for further research, such as (Kong et al 2021, Kumar et al 2022, Veeman et al 2022, Ramadoss et al 2023. Very lately, (Kumar et al 2022) proposed a new chaotic map derived from the logistic map with a dividend + x 1 .…”

“…However, it is very likely to come across a couple of novel chaotic maps presented in the literature which could be one-dimensional (Yosefnezhad Irani et al 2019, Talhaoui and, two-dimensional (Jiang et al (2019), Wang and Ding 2018) and multi-dimensional (Valandar et al 2019). However, all of these papers focus on the application areas of the maps; therefore, they are not fully committed to investigating the nonlinear dynamics, except some mathematical papers like (Ji et al 2018, Ramadoss et al 2023. These works mentioned in this section also verify the necessity of the proposing new and unique maps with detailed analysis.…”