Feedback Loops

266. Feedback Loops

Closed causal chains where an effect comes back to influence its cause. The engine of dynamic behavior in system dynamics.

266.1. Two types

Reinforcing loops (R, positive feedback): the loop amplifies changes. Math: $\cdot 𝑥 = 𝑘 𝑥$ with $𝑘 > 0$ . Solution: exponential growth $𝑥 (𝑡) = 𝑥_{0} 𝑒^{𝑘 𝑡}$ .

Balancing loops (B, negative feedback): the loop resists changes. Math: $\cdot 𝑥 = - 𝑘 𝑥 + 𝑘 𝑥^{*}$ with $𝑘 > 0$ . Solution: exponential approach to target $𝑥^{*}$ with time constant $\frac{1}{𝑘}$ .

266.2. Behavior patterns

Loop structure	Behavior	Example
Pure reinforcing	Exponential growth/collapse	Compound interest, viral spread
Pure balancing	Goal-seeking to steady state	Thermostat, predator-prey equilibrium
R + B (limit)	S-curve (logistic)	Market penetration, learning curves
Two B with delay	Oscillation (damped or sustained)	Inventory cycles, beer game
R + B unstable	Overshoot and collapse	Resource depletion

266.3. First-order behavior

Reinforcing ( $\cdot 𝑥 = 𝑔 𝑥$ ):

𝑥 (𝑡) = 𝑥_{0} 𝑒^{𝑔 𝑡}

Doubling time: $𝑇_{2} = \frac{ln (2)}{𝑔} \approx \frac{0.69}{𝑔}$
Halving time (if $𝑔 < 0$ ): same formula, opposite sign

Balancing ( $\cdot 𝑥 = \frac{𝑥^{*} - 𝑥}{𝜏}$ ):

𝑥 (𝑡) = 𝑥^{*} + (𝑥_{0} - 𝑥^{*}) 𝑒^{- \frac{𝑡}{𝜏}}

Time constant $𝜏$ : time to reach $63 %$ of the way to $𝑥^{*}$
Settling time (within 5%): $\approx 3 𝜏$
Settling time (within 1%): $\approx 5 𝜏$

266.4. Combined loops: limits to growth

Reinforcing growth ( $\cdot 𝑥 = 𝑔 𝑥$ ) plus balancing loop tied to capacity $𝐾$ :

\cdot 𝑥 = 𝑟 𝑥 (1 - \frac{𝑥}{𝐾}) (logistic growth)

Initially R dominates → exponential growth. As $𝑥 \to 𝐾$ , the balancing factor $(1 - \frac{𝑥}{𝐾})$ shrinks $\cdot 𝑥$ . Asymptote: $𝑥 \to 𝐾$ . S-curve.

See Logistic Growth.

266.5. Second-order: oscillation

A loop with delays produces oscillation. Simple example: $\cdot \cdot 𝑥 + 2 𝜁 𝜔 \cdot 𝑥 + 𝜔^{2} 𝑥 = 0$ (damped harmonic oscillator).

Damping $𝜁$	Behavior	Pattern
$0$	Sustained oscillation	sine wave
$0 < 𝜁 < 1$	Damped oscillation	decaying sine
$𝜁 = 1$	Critical damping	smoothest approach
$𝜁 > 1$	Over-damped	no oscillation, slow approach

266.6. Why feedback loops matter

Most “surprising” system behavior comes from feedback:

Bullwhip in supply chains: balancing loops with delays amplify variance
Boom-and-bust cycles: capacity adjustment delays
Population dynamics: birth/death feedbacks
Climate: ice-albedo, methane release reinforcing loops; weathering balancing loops

Without feedback, a system can’t generate dynamic patterns; it’s just open-loop input → output.

266.7. Identifying loops in a system

List variables and how they affect each other
Trace cycles in the diagram
Compute polarity of each cycle (R or B)
Identify dominant loops in each behavior phase

The same model can have R-dominant phase (early growth) and B-dominant phase (saturation).