Existence, Stability, and Control of Glucose-Insulin Dynamics via Caputo-Fabrizio Fractal-Fractional Operators

This study presents a novel numerical framework for simulating glucose-insulin regulatory dynamics using the Caputo-Fabrizio (CF) fractal-fractional operator with both constant and variable fractional orders. The model incorporates an exponential decay kernel to capture memory and hereditary effects in metabolic regulation. A Newton interpolation-based numerical scheme is developed to approximate the CF-FF derivatives, ensuring computational stability and accuracy. For the variable-order formulation, the fractional order $\beta \left(t\right)$ dynamically evolves with time, reflecting physiological variability typically observed during intravenous glucose tolerance tests (IVGTT). Numerical experiments reproduce physiologically realistic glucose-insulin oscillations and demonstrate how feedback control stabilizes chaotic metabolic behavior. The results are based entirely on simulation evidence calibrated within clinically reported parameter ranges, providing conceptual validation rather than direct patient-data comparison. The proposed approach bridges mathematical fractional calculus with biomedical applications, offering new insights for personalized diabetes management and adaptive glucose control strategies.•Fractal-fractional model formulation capturing glucose-insulin memory and adaptation•Stable numerical scheme using Newton interpolation for accurate fractional integration•Linear feedback control applied to regulate chaotic glucose-insulin dynamics•Numerical Methodology for glucose-insulin dynamics. Our investigation of the fractal-fractional glucose-insulin system employs the following analytical framework:•Model Development: We formulate a fractal-fractional-order extension of the minimal glucose insulin model, incorporating an exponential decay type kernel to capture the system's memory effects and anomalous diffusion characteristics inherent in metabolic processes. The model accounts for both insulin-dependent and independent glucose utilization dynamics.•Computational Implementation: We develop a novel numerical solver based on Newton's interpolation polynomials, implementing the Atangana-Seda fractal-fractional derivative formulation. This method provides an efficient computational framework for solving the coupled nonlinear fractional differential equations while maintaining numerical stability across different fractional orders.•The purpose of this section is to define a mathematical model to study the dynamic behavior of glucose-insulin physiology.•With the Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying dissipative.•In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.