A Unified Approach to Stability Analysis and Approximation of Nonlinear Dynamical Systems via Fractional-Order Methods and Machine Learning Surrogates

Project Abstract

In this work, we present a unified framework for stability analysis and high-fidelity approximation of nonlinear dynamical systems by integrating fractional-order calculus with data-driven machine learning surrogates. We investigate the qualitative dynamics of nonlinear systems through fractional differential models, leveraging their intrinsic memory effects and rich stability properties to capture complex behaviors that integer-order models may overlook. Our approach develops a rigorous theoretical foundation for stability criteria in fractional-order systems, including generalized Lyapunov methods, Mittag-Leffler stability, and dissipation inequalities tailored to Caputo and Riemann-Liouville formulations. Complementing the analytical insights, we design robust numerical schemes for simulating fractional dynamics with adaptive time-stepping and spectral accuracy, ensuring computational efficiency for high-dimensional problems. The core contribution lies in constructing machine learning surrogates—neural networks and kernel-based models—that approximate the nonlinear map and its fractional-order dynamics with quantifiable error bounds. We introduce regularized training objectives that encode physical laws, memory kernels, and stability constraints, enabling surrogates to respect inherent system invariants and long-term behavior. To bridge theory and practice, we integrate the surrogates into a hybrid solver that switches between low-order fractional models for rapid exploration and high-fidelity surrogate-augmented simulations for precise predictions near critical regimes such as bifurcations and chaos onset. A comprehensive error analysis establishes convergence rates and stability margins for the hybrid framework under model uncertainty and data noise. The methodology is validated on a suite of benchmark nonlinear systems, including fractional-van der Pol oscillators, fractional chaotic Lorenz systems, and delay-augmented nonlinear maps reformulated in fractional form. We demonstrate that fractional-order models capture memory-driven phenomena such as hysteresis, long-range dependence, and anomalous diffusion more faithfully than their integer-order counterparts, while machine learning surrogates deliver real-time predictions with rigorously bounded errors. The proposed framework enables reliable long-term forecasting, effective control design, and robust parameter estimation for complex dynamical phenomena across engineering, physics, and applied mathematics. Sensitivity analyses reveal critical influences of fractional order and kernel choice on stability regions and attractor structures, guiding practical model selection. The study also discusses computational trade-offs, data requirements, and strategies for transferring models across related systems. Overall, this work advances a cohesive methodology that synergizes fractional calculus and data-driven modeling to enhance stability analysis, accurate approximation, and efficient simulation of nonlinear dynamical systems in contexts where memory effects and nonlinear complexity are paramount.

Project Overview

What This Project Is About

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A hands-on exploration of how changing the way we model dynamic systems and how we predict their behavior can improve stability and approximation. It combines fractional-order modeling, which uses non-integer rates of change, with machine learning surrogates that stand in for complex computations to speed up analysis.

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The Problem It Addresses

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Many nonlinear systems are difficult to analyze with traditional methods, leading to uncertain stability assessments and costly simulations. This project looks for a unified approach that makes stability checks more robust and accurate while speeding up predictions through data-driven surrogates.

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Objectives of the Project

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1. Understand fractional-order models and why they can capture memory effects in dynamics.
2. Explore stable and accurate ways to approximate nonlinear behavior using surrogates.
3. Develop a framework that combines both approaches for reliable stability analysis.
4. Evaluate the framework on representative nonlinear systems.

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What You Will Do Step by Step

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1. Learn the basics of fractional calculus and common nonlinear models. 2. Build or adapt simple fractional-order models of selected systems. 3. Create machine learning surrogates trained on simulation data. 4. Integrate the two methods into a unified analysis workflow. 5. Test stability predictions against full simulations. 6. Analyze accuracy, efficiency, and robustness. 7. Document results and potential real-world applications.

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Expected Outcome

Delivered framework and a set of guidelines showing when fractional-order models plus surrogates improve stability assessment, with demonstrations on test systems and insights into practical use and limitations.