Numerical Techniques for Solving Fractional Glucose–Insulin Regulatory Systems

Diabetes mellitus represents a growing global health crisis, marked by complex glucose–insulin regulatory dynamics that exhibit stability transitions and chaotic behaviors. This study employs fractional-order calculus to model these dynamics, capturing critical memory effects inherent in biological systems. The ARA-residual power series method (ARA-RPSM) is utilized to derive approximate analytical solutions for the fractional-order glucose–insulin system, which are validated through numerical simulations. The analysis reveals critical stability thresholds, bifurcations, and transitions to chaos, emphasizing the influence of fractional-order parameters and physiological factors. Stability and chaos analyses, supported by Lyapunov exponents and bifurcation diagrams, highlight the system’s sensitivity to parameter variations and initial conditions. These findings underscore the potential of fractional-order modeling in diabetes research, offering actionable insights for stabilizing glucose–insulin interactions and managing chaotic fluctuations. The study further demonstrates the computational efficiency of ARA-RPSM in exploring fractional-order systems, paving the way for advanced therapeutic strategies in diabetes management.