266. Second-Order
A two-stock system with delay between input and output. The classic oscillator structure in system dynamics.
266.1. Generic equation
where:
- : natural frequency (radians/time)
- : damping ratio (dimensionless)
- : target / equilibrium
266.2. Behavior by
| Type | Behavior | |
|---|---|---|
| Undamped | Pure sinusoidal oscillation, no decay | |
| Underdamped | Decaying oscillation | |
| Critically damped | Smoothest non-oscillatory approach | |
| Overdamped | Slow non-oscillatory approach |
266.3. Damped frequency and period
For underdamped systems:
266.4. Where 2nd-order systems arise
Whenever a balancing loop has a delay equal to (or exceeding) the loop’s time constant:
- Inventory control with order delay: targeting inventory level via delayed orders
- Production-inventory: production responds to inventory gap with lead time
- Workforce: hiring decisions based on delayed performance signals
- Predator-prey: population responds to delayed food availability
266.5. Beer game equations as 2nd-order
In the Sterman beer game model, each echelon’s order rate depends on:
- Current inventory gap
- Delayed perception of demand
- Delayed shipments from upstream
Combining gives a 2nd-order ODE in inventory and orders → oscillation visible in real beer-game data and field supply chains.
266.6. Stability via Routh-Hurwitz
For a 2nd-order linear system , stable iff all have the same sign (Routh-Hurwitz criterion). For SD systems, typically; need (damping) and (restoring force) for stability.
Linearization around a fixed point gives this form locally; eigenvalues of the Jacobian determine stability.
266.7. See also
- Feedback Loops — 1st-order foundations
- Delays — what creates oscillation
- Phase Plane — visualization
- Beer Game — application