A Lyapunov–Cycle Framework for Positive Nonlinear Systems with Third-Order Curvature Dynamics

We introduce a Lyapunov–cycle architecture for nonlinear dynamical systems evolving on the positive orthant and coupled to a third-order curvature subsystem. The stability functional S(x) = -∑_{i=1}^n w_i ln x_i + ∑_{i=1}^n α_i x_i, w_i>0, α_i≥0 satisfies Ṡ(x(t)) = B(x(t),z(t)) ≥ 0 along trajectories, rendering every superlevel set Σ_c := {x : S(x) ≥ c} forward-invariant. A composite Lyapunov function W(x,z) = S(x) + L(z) with a globally exponentially stable third-order curvature dynamics produces forward-invariant sublevel tubes T_c := {W ≤ c}. A multiplicative mirror-ascent update x⁺ = x ⊙ exp(η ∇S(x)), η>0 strictly increases S while preserving positivity. The framework is instantiated on a deterrence-stability model with cross-escalation probability P_cross = T_D P_D L e^{-α D}, yielding a complete, mathematically rigorous stability tube that drives the system to the risk-free regime.