A Framework for Dynamic Stability Analysis of Nonlinear Differential Equations

Thesis Abstract

Nonlinear differential equations are fundamental to modeling complex dynamical systems across various scientific and engineering disciplines; however, their inherently intricate behavior poses significant challenges to stability analysis, especially in ensuring reliable predictions of system behavior under varying conditions. This study aims to develop a comprehensive framework for the dynamic stability analysis of nonlinear differential equations, focusing on advancing theoretical understanding and providing practical analytical tools for researchers and practitioners. The primary objectives include (1) synthesizing existing stability criteria and embedding them within an integrated analytical model, (2) identifying critical parameters influencing stability through sensitivity analysis, and (3) validating the framework's robustness across diverse nonlinear systems representative of physical, biological, and engineering contexts. Employing a mixed-methods research design, the study integrates theoretical development with empirical validation. The theoretical component involves an extensive review of current stability theories, notably Lyapunov’s direct method and the bifurcation theory, which serve as the foundational frameworks for the proposed model. The empirical component encompasses a systematic examination of fifty nonlinear differential equations sourced from contemporary literature, encompassing systems such as ecological models, electrical circuits, and mechanical oscillators. The population of these models was compiled from reputable journals and online repositories, ensuring diverse representation. A purposive sampling technique was employed to select systems with well-documented behaviors and known stability properties, facilitating targeted validation and comparative analysis. Data collection involved extracting analytical parameters, bifurcation points, and stability conditions from the selected models, utilizing advanced symbolic computation tools such as MATLAB’s Symbolic Math Toolbox and AUTO-07p for bifurcation analysis. The validity of theoretical constructs was ensured through peer-reviewed literature corroboration, while reliability was established by cross-verifying stability outcomes obtained via the proposed framework against established analytical solutions for benchmark models. The analysis mainly employed numerical bifurcation analysis, Lyapunov function construction, and sensitivity analysis to evaluate the stability implications of parameter variations. These techniques enabled the development of a generalized stability assessment algorithm, integrated within a computational toolkit aimed at automating stability diagnosis. Expected findings include a cohesive stability assessment framework that incorporates multiple criteria—Lyapunov stability, bifurcation analysis, and amplitude-phase conditions—under a unified analytical schema. The framework is anticipated to enhance predictive accuracy for complex dynamical behaviors, such as chaos and multi-stability, across a broad spectrum of nonlinear systems. Moreover, the sensitivity analysis component is expected to identify key parameters that predominantly govern stability thresholds, thus informing control strategies for system stabilization. This research’s primary contribution lies in bridging theoretical stability criteria with practical analytical procedures, resulting in an accessible, scalable framework adaptable to various nonlinear systems. It extends existing theories by formalizing an integrative approach that combines Lyapunov methods with bifurcation analysis within a computational environment, enriching the theoretical toolkit available for dynamic system analysis. Additionally, the study offers a validated computational model that enables early stability diagnostics, which can significantly benefit multidisciplinary fields such as control engineering, ecology, and applied physics. Conclusively, the study recommends integrating the developed framework into existing simulation platforms to foster broader application. Future research should explore extending the framework to stochastic differential equations and high-dimensional systems, potentially enhancing its utility in modeling real-world phenomena characterized by uncertainty and complex interactions. The findings are expected to significantly advance the capacity for stability analysis in nonlinear dynamical systems, supporting the development of more resilient and adaptive systems across scientific domains.