An Integrated Differential-Equation Framework for Stability, Sensitivity, and Computational Analysis of Complex Dynamical Systems
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Abstract
Complex dynamical systems in engineering, biology, economics, environmental science, and social processes are characterized by nonlinear interactions, parameter sensitivity, and evolving behavior that descriptive analysis alone cannot adequately examine. This study develops an integrated mathematical and computational framework to model and analyze complex dynamical systems using differential equations. The framework combines state-space formulation, parameter representation, equilibrium analysis, Jacobian linearization, eigenvalue-based stability classification, Lyapunov stability criteria, numerical simulation, parameter sensitivity analysis, and statistical model validation. Without empirical data, we evaluate the framework using simulated benchmark models, including logistic growth and a two-variable predator–prey system with logistic prey growth. Numerical experiments examine temporal trajectories, phase-space behavior, equilibrium convergence, and predictive accuracy. The results confirm the analytical stability properties of both benchmark systems: the logistic model converges toward its carrying-capacity equilibrium, while the two-variable system approaches a locally asymptotically stable coexistence equilibrium through damped oscillations. Validation against pseudo-observed data produced an MAE of 1.9446, RMSE of 2.4552, R² of 0.9917, and MAPE of 3.1017%, demonstrating strong consistency between the simulated observations and model predictions. Overall, the study provides a reproducible framework that integrates analytical stability theory, numerical simulation, sensitivity analysis, and statistical validation for systematically investigating nonlinear dynamical systems.
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