Structural Transition Dynamics of Constrained Complex Systems: A Hybrid Dynamical Framework Based on Structural Potential and Critical Reconfiguration

Complex systems frequently undergo gradual structural distortion followed by abrupt topological reorganization, a phenomenon insufficiently captured by classical dynamical systems with time-invariant governing vector fields. This paper establishes Structural Transition Dynamics (STD), a hybrid dynamical modeling framework tailored for constrained complex systems, built upon three core primitives: a structural potential function, endogenous critical transition criteria, and discrete state reconfiguration operators. We formalize the target system as a constrained hybrid tuple composed of compact admissible state space, continuous evolution law, structural potential manifold, constant critical threshold, and jump reconfiguration mechanism. Continuous dynamics govern slow accumulation or intrinsic relaxation of structural misalignment; once structural potential exceeds the critical threshold, the system triggers an instantaneous discrete state switch. Under standard compactness, Lipschitz regularity and smoothness assumptions, we rigorously derive finite-time critical triggering theorems and Lyapunov stability guarantees for post-transition configurations. Distinct from traditional modeling paradigms limited to fixed structural topology, the proposed framework embeds structural reorganization as an endogenous, state-driven dynamic behavior. This formalism does not supersede existing nonlinear dynamics or thermodynamic theories; instead, it supplies an auxiliary mathematical dimension to analyze constrained systems with evolving internal topological configurations. This study lays a self-consistent general theoretical foundation for hybrid structural transition dynamics, offering a unified mathematical basis for subsequent research on complex networks, adaptive engineering systems, nonlinear bifurcation analysis, and computational complex system simulation.