Coherence Collapse in a Coupled Stress-Recoherence Dynamical System

ABSTRACT We study the coupled two-dimensional dynamical system governing a coherence variable C (t) in -1, +1 and an accumulated load variable L (t) >= 0: dC/dt = -alpha*sigma* (1+C) + beta*R (L) * (1-C) ; dL/dt = sigma - R (L), with R (L) = R₀*exp (-kL) and parameters alpha, beta, R₀, k, sigma > 0. We establish the following results, each completely proved: Proposition 0 (Well-posedness): Unique global solutions exist for all bounded measurable stress inputs sigma (t), with C (t) in -1, +1 for all time. Lemma 1 (Equilibrium): The unique equilibrium C* (L) = rho-1/rho+1 where rho = beta*R (L) /alpha*sigma is locally asymptotically stable with rate lambda = alpha*sigma + beta*R (L). Lemma 2 (Lyapunov): The complexity maintenance cost function Phi (C) = alpha*sigma* (1+C) /beta* (1-C) characterises coherence dynamics: dV/dt Phi (C), where V (C) = (1-C) ²/2. Proposition 1 (Bifurcation): The equilibrium sign changes at rho = 1, with quasi-static tracking established via Fenichel's theorem and bifurcation delay characterised for rapid parameter variation. Theorem 1 (Collapse): For persistent load L (t) > Lcritical, C (t) converges monotonically to Cᵢnfinity 0 as epsilon -> 0 (small-noise limit, Freidlin-Wentzell) with rate exp (-I*/epsilon²). For fixed epsilon, Pᵢnternal <= 2*exp (-2*mu₀*Delta/epsilon²), a positive constant independent of L₀. MSC2020: 34D20, 34D23, 34C23, 60H10, 92B05 Keywords: Lyapunov stability, coupled ODE, coherence collapse, complexity maintenance cost, parameter bifurcation, omega-limit sets, Fenichel theorem, Freidlin-Wentzell, quasi-static approximation, external modulation necessity