Coupled Pressure–Capacity Dynamics and Bifurcation Cascades in a Piecewise-Smooth Model of Standardization Pressure

We extend a one-dimensional piecewise-smooth model of standardization pressure into a two-dimensional coupled dynamics, where the standardization pressure P and the intermediate-layer capacity M evolve interactively. When the pressure–capacity coupling coefficient χ vanishes, the pressure equation recovers the original piecewise-smooth pressure model, while the capacity variable may remain driven by pressure through η. Full dynamical decoupling occurs only when both χ and η vanish. We rigorously establish the forward invariance of the state space [0,1]^2 by introducing boundary-modulating factors. Through an exact algebraic reduction, we eliminate the capacity variable and obtain a scalar equilibrium equation. We analyze the root counts and local stability of equilibria within each smooth branch, and provide a complete inventory for the specified parameter set. This includes the derivation of a transcritical bifurcation at the reference point and a non-degenerate saddle-node bifurcation induced by a narrow parameter window. By calculating the exact one-sided vector fields, we classify the existence of boundary-collision equilibria under the right-continuous convention. Finally, we explicitly distinguish three analytical tiers: the algebraic equilibrium equations, the local stability within smooth branches, and the non-smooth switching dynamics. No empirical claims are made; all results are mathematical and conditional on the stated parameters.