ASYMPTOTIC ANALYSIS OF SUSTAINABLE EDUCATIONAL QUALITY: STABILITY AND CONVERGENCE IN A NONLINEAR DELAY MODEL

This study introduces a third-order nonlinear delay differential equation (DDE) model designed to simulate the dynamics of educational quality, aligning specifically with the goals of Sustainable Development Goal 4 (SDG 4). The model is structured to incorporate time lags which accurately reflect real-world systemic delays in areas such as policy implementation, resource allocation, and feedback from learning outcomes. Educational quality is the primary dependent variable, with the DDE's delay components capturing the inertia inherent in complex educational systems. Using the rigorous Lyapunov's direct method, augmented with integral terms, the research theoretically proves the uniform asymptotic stability and convergence of solution pairs under bounded conditions. This theoretical finding is crucial, as it ensures that initial disparities or differences between various educational states will asymptotically diminish over time, converging toward a stable, common equilibrium. The study validates these theoretical results through extensive numerical simulations performed in Mathematica. These simulations, which include phase portraits and qualitative analysis across varying delay magnitudes, confirm that small time delays are critical for promoting stable, bounded solutions and achieving rapid convergence to the desired quality standard.