System Dynamics & Control Theory: Dynamic Systems Formulation of the Odero Persistence Framework

System Dynamics & Control Theory develops a dynamic systems and state-space formulation of persistence based on the Odero Axiom and the Persistence Ratio framework. Building upon the persistence condition C(I)E > D, the work extends persistence analysis from static measurement to dynamic trajectory modeling through time. The framework is centered on the Persistence Index: Ω = C(I)E / D which measures persistence capacity relative to decay. Persistence dynamics are governed by: Omega Velocity (ω) and Omega Acceleration (α): ω = Ω[(ḊC(I)/C(I)) + (ḊE/E) − (ḊD/D)] α = d²Ω/dt² The complete persistence state of a system is represented by the state vector: Ψ(t) = [Ω, ω, α, κ] This publication introduces: A state-space representation of persistence dynamics. Omega Acceleration (α) as a second-order persistence metric. The Persistence State Vector Ψ(t). The Odero Control-Phase Simplex. Governance phase classifications (Tyranny, Structural, Paralysis). Finite-time collapse prediction. Phase-boundary hysteresis effects. Lagrangian persistence optimization. Numerical persistence trajectory modeling. The framework further introduces the Steering-to-Engine Ratio: R = C(I)/Ê and defines three governance regions: Tyranny (R < 0.5) Structural Region (0.5 ≤ R ≤ 2.0) Paralysis (R > 2.0) for analyzing organizational balance, control efficiency, and systemic stability. Relationship to Previous Work: The Persistence Ratio (Index) introduced persistence measurement through Ω and ω. This work extends those foundations into a full state-space framework for modeling persistence, recovery, deterioration, governance dynamics, and collapse trajectories over time