Structural Stability and Regime Classification in Discrete-Time State–Event–Response Systems Through Induced Transition Topology

This paper develops a finite-state mathematical framework for structural stability and regime classification in discrete-time state–event–response systems whose effective transition structure is generated endogenously by state-dependent response rules. Unlike classical structural stability theory, which focuses on qualitative persistence in smooth dynamical systems, and unlike Markov-chain analysis, which typically assumes a fixed transition kernel, the proposed framework treats the transition graph as an induced object. The model specifies a finite state space, an event-generation law, an elasticity-dependent attenuation function, and a deterministic transition mapping. Structural regimes are classified by adjacency relations, communicating components, absorbing organization, and long-run occupancy support. A Monte Carlo verification layer is used only to examine whether the analytically defined topological regimes are visible in finite-sample occupancy signatures. The results indicate that, within the finite-state setting considered here, admissible disturbance scaling changes traversal frequency without changing graph identity, whereas elasticity variation can activate or deactivate effective edges and thereby generate structurally distinct regimes.