Degenerate Fold and Local Stable Branches in a Capacity-Constrained Piecewise-Smooth Dynamical System

This paper investigates a three-dimensional piecewise-smooth dynamical system characterized by a transverse variable and a hard capacity constraint. Building upon the original (P)-(\Gamma)-(M) system, we introduce the realized input Q = \min\{q_j, \kappa M\}, where q_j \ge 0 represents the given external input level and \kappa > 0 is the capacity coefficient. This extension recovers the original system in the capacity-abundant regime while introducing state-dependent feedback in the capacity-limited regime.

On the invariant subspace \Gamma=0, the equilibrium problem is reduced to a scalar equation:

L(M) = \min\{q_j, \kappa M\}.

We distinguish between the capacity-abundant and capacity-limited branches, establishing the linear stability criteria for each smooth region. Furthermore, we analyze the degenerate switching point satisfying L(M_0) = q_0 and L'(M_0) = \kappa_0. Under appropriate non-degenerate, dissipative, and transverse stability conditions, we prove that a square-root type local branch emerges on the capacity-limited side. One of these branches strictly satisfies the capacity feasibility constraint and is locally asymptotically stable in the complete three-dimensional system.

The conclusions are strictly local, avoiding claims of global uniqueness, global attraction, or general non-smooth stability classification on the switching manifold. We further verify a specific parameter instance using exact rational arithmetic, clearly distinguishing between the general theorem, the parameter instance, and numerical evidence.