Analyzing fractional glucose-insulin dynamics using Laplace residual power series methods via the Caputo operator: stability and chaotic behavior

Abstract

Abstract Background The dynamics of glucose-insulin regulation are inherently complex, influenced by delayed responses, feedback mechanisms, and long-term memory effects. Traditional integer-order models often fail to capture these nuances, leading to the adoption of fractional-order models using Caputo derivatives. This study applies the Laplace residual power series method (LRPSM) to explore the glucose-insulin regulatory system’s stability, oscillatory behaviors, and chaotic transitions. Results Morphologically, the fractional-order glucose-insulin regulatory system revealed transitions between stability, oscillations, and chaos. Key system behaviors were characterized using Lyapunov exponents, bifurcation diagrams, and phase portraits. Numerical simulations validated the effectiveness of LRPSM in capturing essential dynamics, including sensitivity to parameters such as insulin sensitivity and glucose uptake rates. The chaotic behaviors observed emphasize the system’s sensitivity to initial conditions and fractional order. Conclusion This study highlights the utility of LRPSM in modeling fractional-order biological systems, offering significant advancements in understanding diabetes pathophysiology. The findings pave the way for designing glycemic control strategies and exploring optimized interventions for diabetes management. Future research could integrate additional physiological parameters and explore real-time applications to enhance glycemic control.