A Unified Framework for Fractional-Order Dynamical System Stability Analysis

Thesis Abstract

Fractional-order dynamical systems (FODS) offer a versatile modeling framework for intrinsic memory effects and anomalous diffusion across engineering, physics, and applied mathematics. The study addresses the challenge of establishing a unified stability assessment framework for FODS that integrates structural, spectral, and fractional calculus-based criteria to derive robust, implementable conditions applicable to a broad class of nonlinear systems. The aim is to develop a coherent theoretical framework that (i) unifies Caputo, Riemann–Liouville, and Grünwald–Letnikov formulations under a common stability criterion, (ii) synthesizes Lyapunov–like functionals tailored to fractional dynamics, and (iii) provides computationally tractable verification procedures compatible with data-driven modeling. Specific objectives include (1) formulating a generalized Routh–Hurwitz–type criterion adapted to fractional orders and multi-stability; (2) deriving fractional small-gain and Lyapunov-based sufficient conditions for global and local stability; (3) constructing a modular stability analysis toolkit that accommodates parameter uncertainty and perturbations; (4) validating the framework on a suite of benchmark FODS models including fractional-order Van der Pol, fractional-order Lorenz, and fractional-order neural oscillators; and (5) demonstrating applicability to real-world systems in control engineering and biophysical modeling. The methodology adopts a theory-driven, mixed-methods design that combines analytical derivations with numerical validation. The population of interest comprises nonlinear FODS models in continuous time across three representative domains (a) autonomous oscillators, (b) chaotic fractional-order systems, and (c) control-implemented fractional systems with actuator/measurement noise. A purposive sample of 12 well-characterized models and 6 experimentally-inspired or data-validated systems is employed. Analytical work proceeds in two stages first, a symbolic derivation of stability conditions using spectral mapping theorems for fractional operators and a generalized Lyapunov–Krasovskii functional approach; second, a computational stage implementing the stability tests using MATLAB and the FOAL toolkit, with code validated against analytical benchmarks. Data collection relies on simulated time-series generated from canonical fractional differential equations and, where available, experimental measurements from fractional-order circuits and neurobiological models. Instruments include symbolic computation routines for deriving characteristic equation bounds, numerical solvers for fractional differential equations (Adams–Bashforth–Mirkovitch schemes), and stability verification modules that compute fractional-order sector conditions and Lyapunov functionals. Validity and reliability are ensured through cross-validation against known stability results in the integer-order limit, convergence analyses for discretized fractional schemes, and sensitivity analyses across order ? ? (0,1] and parameter uncertainty bands. Method of analysis integrates analytical proofs with numerical experiments, including Monte Carlo simulations (n = 200 trials) to assess robustness, and regression-based calibration to map observed stability regions to system parameters. The key expected findings include (i) a unified set of sufficient conditions that reduce to classical criteria as ? ? 1, (ii) explicit Lyapunov functionals for common fractional-order models that guarantee global asymptotic stability under realistic perturbations, (iii) a modular stability toolkit with step-by-step verification procedures, and (iv) practical guidelines for selecting fractional orders and controller gains to ensure desired stability margins. The study contributes to knowledge by bridging fractional calculus theory with practical stability verification, delivering a universal framework that supports both theoretical analysis and data-informed modeling of FODS. It advances the state of the art by providing a coherent, extensible methodology that unifies disparate stability notions, facilitates rigorous design in control applications, and informs model validation in complex systems exhibiting memory effects. The main conclusion expects that the proposed unified framework delivers consistent, order-aware stability criteria applicable across diverse FODS, with recommendations for practitioners to adopt the modular toolkit for model development, numerical verification, and controller synthesis in engineering and scientific contexts. Recommendations include extending the framework to stochastic fractional dynamics, integrating adaptive order estimation, and developing a standardized benchmark repository of fractional-order models for ongoing methodological refinement.