Regime-Limited Dynamical Systems: Theory and Reduction Criteria for Singular ODEs

This work presents a general mathematical framework for Regime-Limited Dynamical Systems (RLDS), substantially extending the preliminary results presented in our earlier work (DOI: 10.5281/zenodo.18274782), where a specific system in the unimodal case was analyzed. The current paper develops a complete theory covering both unimodal and monotone geometric cases, introduces explicit reduction criteria for systematic application to external models, and establishes structural stability under perturbations. We introduce and rigorously analyze a class of one-dimensional autonomous ordinary differential equations on bounded intervals that exhibit regime-limited dynamics: long-time evolution constrained by a globally attracting equilibrium acting as a robust limiter of the system state. This class—termed Regime-Limited Dynamical Systems (RLDS)—is characterized by competition between quadratic dissipation and a singularly modulated restoring term, optionally gated by a smooth switching profile. The core analytical tool is an auxiliary function H whose geometry yields a constructive characterization of equilibria as intersections with a single regime parameter κ = β/α. We define two key thresholds: κc = sup H (the critical supremum) and κbdy = H(xc⁻) (the boundary limit). The theory accommodates both unimodal H (with interior maximum) and monotone H (supremum at boundary). Under minimal regularity assumptions (C¹ smoothness), we establish: (i) existence of equilibria, (ii) uniqueness for κ < κc, (iii) global asymptotic stability, and (iv) structural stability with respect to perturbations. A second contribution is methodological: we provide explicit reduction criteria (Conditions C1–C4) for embedding external models into the RLDS framework. The theory is illustrated with a concrete realization (System A) representing the monotone case, where H is strictly increasing and κc = κbdy, yielding a unique global attractor at x* ≈ 0.231.