Dynamical properties of a novel one dimensional chaotic map

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Mathematical Modeling

Background:

• The logistic map is a widely used model for demonstrating chaotic behavior.
• Existing chaotic maps have limitations in their range of stability and chaotic dynamics.
• Novel chaotic systems are needed for improved performance in various applications.

Purpose of the Study:

• To introduce and analyze a novel one-dimensional chaotic map, K(x).
• To investigate the dynamical properties, stability, and chaotic behavior of the proposed map.
• To compare the performance of the new map with the traditional logistic map.

Main Methods:

• Analysis of fixed points, attracting points, and repelling points.
• Utilizing bifurcation diagrams, cobweb representations, and time series analysis.
• Computation of maximal Lyapunov exponent, entropy, and probability distribution.

Main Results:

• The proposed K(x) map exhibits complex dynamical properties and chaotic behavior.
• The K(x) map demonstrates a larger range of stability and chaos compared to the logistic map.
• Entropy and probability distribution analysis show distinct characteristics compared to the logistic map.

Conclusions:

• The novel K(x) chaotic map is a promising alternative to the logistic map.
• Its extended range of stability and chaos suggests potential for enhanced performance in applications.
• Further research can explore its utility in cryptography, secure communication, and other fields.