Ismail’s Contemporary Pointwise Stationary Fluid Approximation Theory: Discovering the Unknown Transitions Between Stability, Traffic Intensity, and Chaos of the Non-Stationary

This research investigates the unexplored domain of negative parameters within dynamic systems, focusing on their influence across stability, traffic intensity, and chaotic phases. Using an iterative computational framework, we examine the sigma function (σ) to characterize system responses under varying conditions of and . Visualizations and insights demonstrate transitions between phases for a first time-ever exploration for negative values of the number of sets of phases, namely k, with novel findings extending foundational studies. This work establishes a baseline for further explorations of negative parameter spaces in complex systems. It is to be noted that the current work provides new contributions to Ismail’s Contemporary Pointwise Stationary Fluid Approximation Theory by offering a comprehensive computational framework for exploring dynamic system behaviors across varying parameters.