New Approaches to Generalized Logistic Equation with Bifurcation Graph Generation Tool

This paper propose two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic equation and the logistic equation of arbitrary order respectively. It demonstrate the impact of chaos theory by integrating it with logistics equations and reveal how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. As well, this work presents BifDraw – a Python program for making bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system's response to the changes in the conditions. The research gives a pivotal role to the logistic equation's place in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigates dynamics nature of the equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic equations within chaos theory. This study not only deepens our comprehension of logistic equations and chaos theory but also introduces practical tools for visualizing and analyzing their behaviors.