Unified Data-Driven Framework for Hopf Bifurcation Analysis, Machine Learning, Phase Space Characterization, and Bifurcation-Aware Optimal Control of Nonlinear Dynamical Systems

Abstract

Lakshmi. N. Sridhar

Nonlinear dynamical systems frequently exhibit multiple steady states, limit points, and self-sustained oscillations that can significantly affect process stability and operational performance. This study presents an integrated computational framework that combines nonlinear bifurcation analysis, data-driven machine learning, phase-space characterization, and dynamic optimal control to analyze and regulate nonlinear engineering systems operating near critical instability regions. Numerical continuation is first employed to identify the principal bifurcation structure and locate the Hopf bifurcation point together with the associated limit point. A simulation-based dataset is subsequently generated by systematically varying the bifurcation parameter in the vicinity of the critical operating region. Statistical features extracted from the resulting dynamic trajectories are used to train a supervised machine-learning classifier to distinguish between steady-state and oscillatory operating regimes. Phase-space analysis is performed to validate the identified dynamic behavior and to confirm the existence of stable limit cycles associated with the Hopf bifurcation. Finally, dynamic optimal control problems are formulated within Pyomo.DAE framework and solved using IPOPT. Both conventional and bifurcation-aware optimal control formulations are investigated to evaluate the influence of nonlinear stability information on the resulting optimal trajectories. The results demonstrate that the proposed methodology provides an effective framework for integrating nonlinear dynamics, machine learning, and optimization, offering a computationally efficient approach for stability assessment and control of complex nonlinear engineering systems.

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