Flow-Coherence Dynamics in Open Systems - Dissipative Structures, Hopf Bifurcation, and Aptadinamic Viability

We present a three-dimensional nonlinear dynamical framework for open systems traversed by organized energy or informational flow. The system is governed by four macroscopic variables — organized flow Φ, structural coherence C, internal friction γ, and effective entropy production Σ — coupled through an autocatalytic capture function α(C) = α₀ + α₁C that models structural self-amplification. Elevating γ to an independent state variable with inertial rate ω eliminates implicit transcendental dependencies and yields a closed autonomous system in ℝ³ amenable to full analytical treatment. We derive the complete Jacobian at equilibrium, compute the Routh–Hurwitz coefficients p, q, r explicitly in terms of all system parameters, and establish four distinct dynamic regimes: subcritical dissolution, stable thermodynamic branch, metastable limit cycle, and entropic collapse. The Hopf bifurcation emerges at the exact parametric threshold pq = r, producing a limit cycle whose topological signature — Lyapunov spectrum (0, L₂ 0 formalizes systemic growth as increasing structural throughput. The framework connects rigorously to Prigogine's dissipative structures theory and is identified as a concrete physical realization of the Aptadinamia framework for structural viability, establishing a formal bridge between nonequilibrium thermodynamics and non-Markovian viability theory.