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Abstract <p>Diabetes mellitus represents a growing global health crisis,marked by complex glucose–insulin regulatory dynamics thatexhibit stability transitions and chaotic behaviors. This studyemploys 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 toderive approximate analytical solutions for the fractional-orderglucose–insulin system, which are validated through numericalsimulations. The analysis reveals critical stability thresholds,bifurcations, and transitions to chaos, emphasizing the influenceof fractional-order parameters and physiological factors.Stability and chaos analyses, supported by Lyapunov exponents andbifurcation diagrams, highlight the system’s sensitivity toparameter variations and initial conditions. These findingsunderscore the potential of fractional-order modeling in diabetesresearch, offering actionable insights for stabilizingglucose–insulin interactions and managing chaotic fluctuations.The study further demonstrates the computational efficiency ofARA-RPSM in exploring fractional-order systems, paving the way foradvanced therapeutic strategies in diabetes management.</p>

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Numerical Techniques for Solving Fractional Glucose–Insulin Regulatory Systems

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • Sayed Saber,
  • Abdullah Alahmari,
  • Alshaikh A. Shokeralla,
  • Fathelrhman EL Guma

摘要

Abstract

Diabetes mellitus represents a growing global health crisis,marked by complex glucose–insulin regulatory dynamics thatexhibit stability transitions and chaotic behaviors. This studyemploys 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 toderive approximate analytical solutions for the fractional-orderglucose–insulin system, which are validated through numericalsimulations. The analysis reveals critical stability thresholds,bifurcations, and transitions to chaos, emphasizing the influenceof fractional-order parameters and physiological factors.Stability and chaos analyses, supported by Lyapunov exponents andbifurcation diagrams, highlight the system’s sensitivity toparameter variations and initial conditions. These findingsunderscore the potential of fractional-order modeling in diabetesresearch, offering actionable insights for stabilizingglucose–insulin interactions and managing chaotic fluctuations.The study further demonstrates the computational efficiency ofARA-RPSM in exploring fractional-order systems, paving the way foradvanced therapeutic strategies in diabetes management.