Second-Order

268. Second-Order

A two-stock system with delay between input and output. The classic oscillator structure in system dynamics.

268.1. Generic equation

\cdot \cdot 𝑥 + 2 𝜁 𝜔_{𝑛} \cdot 𝑥 + 𝜔_{𝑛}^{2} 𝑥 = 𝜔_{𝑛}^{2} 𝑥^{*}

where:

$𝜔_{𝑛}$ : natural frequency (radians/time)
$𝜁$ : damping ratio (dimensionless)
$𝑥^{*}$ : target / equilibrium

268.2. Behavior by $𝜁$

$𝜁$	Type	Behavior
$0$	Undamped	Pure sinusoidal oscillation, no decay
$0 < 𝜁 < 1$	Underdamped	Decaying oscillation
$𝜁 = 1$	Critically damped	Smoothest non-oscillatory approach
$𝜁 > 1$	Overdamped	Slow non-oscillatory approach

268.3. Damped frequency and period

For underdamped systems:

𝜔_{𝑑} = 𝜔_{𝑛} \sqrt{1 - 𝜁^{2}} (damped frequency)

𝑇_{𝑑} = 2 \frac{𝜋}{𝜔_{𝑑}} (period of oscillation)

268.4. 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

268.5. 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.

268.6. Stability via Routh-Hurwitz

For a 2nd-order linear system $𝑎 \cdot \cdot 𝑥 + 𝑏 \cdot 𝑥 + 𝑐 𝑥 = 0$ , stable iff $𝑎, 𝑏, 𝑐$ all have the same sign (Routh-Hurwitz criterion). For SD systems, $𝑎 > 0$ typically; need $𝑏 > 0$ (damping) and $𝑐 > 0$ (restoring force) for stability.

Linearization around a fixed point gives this form locally; eigenvalues of the Jacobian determine stability.

268.7. See also

Feedback Loops — 1st-order foundations
Delays — what creates oscillation
Phase Plane — visualization
Beer Game — application