Investigating the dynamics of generalized discrete logistic map

Abstract

In recent years, conventional logistic maps have been applied across various fields including modeling and security, owing to their versatility and utility. However, their reliance on a single modifiable parameter limits their adaptability. This paper aims to explore generalized logistic maps with arbitrary powers, which offer greater flexibility compared to the standard logistic map. By introducing additional parameters in the form of arbitrary powers, these maps exhibit increased degrees of freedom, thus expanding their applicability across a wider spectrum of scenarios. Consequently, the conventional logistic map emerges as a specific instance within the proposed framework. The inclusion of arbitrary powers enriches system dynamics, enabling a more nuanced exploration of system behavior in diverse contexts. Through a series of illustrations, this study investigates the influence of arbitrary powers and equation parameters on equilibrium points, their positions, stability conditions, basin of attraction, and bifurcation diagrams, including the emergence of chaotic behavior.