Numerical analysis and stability of a fractional glucose–insulin regulatory system using the laplace residual power series method incorporating the Atangana–Baleanu derivative

This paper investigates the dynamics of a fractional-order glucose–insulin regulatory system using an enhanced Laplace residual power series method incorporating the Atangana–Baleanu derivative. The Atangana–Baleanu derivative, characterized by its nonlocal and nonsingular kernel, offers a more accurate representation of memory and hereditary effects in biological systems compared to traditional derivatives. By employing the enhanced Laplace residual power series method, the study achieves precise numerical solutions and captures the complex behaviors of the system, including stability, bifurcations, and chaotic oscillations. Stability analysis through Lyapunov exponents and bifurcation diagrams reveals the intricate transitions between stable, oscillatory, and chaotic regimes, influenced by fractional-order dynamics and parameter variations. The proposed method demonstrates significant improvements in computational efficiency and solution accuracy, particularly for fractional-order systems with complex interactions. This work highlights the potential of fractional calculus, especially the Atangana–Baleanu framework, in advancing our understanding of glucose–insulin regulation and designing effective strategies for diabetes management.