Topological Collapse and Critical Slowing Down in Resource-Constrained Dissipative Systems

This preprint investigates a minimal class of two-dimensional dissipative systems in which an error-like state variable is nonlinearly coupled to an adaptive reserve that is endogenously eroded by system activity. Using nondimensionalization, nullcline analysis, numerical continuation, and spectral examination of the Jacobian, we identify an interior saddle-node (fold) bifurcation that governs the loss of a functional equilibrium under increasing external forcing. As the fold is approached, the dominant eigenvalue of the linearized system approaches zero, leading to spectral-gap closure and critical slowing down, with a divergence of recovery times. These results provide a conservative, topologically grounded framework for detecting proximity to abrupt state shifts in constrained dissipative flows. The manuscript does not claim a codimension-2 cusp; it focuses specifically on the fold transition and its spectral precursors as a practical early-warning diagnostic in resource-limited nonlinear systems. Nonlinear Dynamics Fold Bifurcation Critical Slowing Down Dissipative Systems Saddle-Node Bifurcation Early Warning Signals Spectral Gap Closure Resource-Depleting Systems