Feedback Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Iterate the map xn+1=0.5xn+3x_{n+1} = 0.5x_n + 3 starting from x0=10x_0 = 10. Compute x1,x2,x3x_1, x_2, x_3 and predict the long-run value.

Solution

  1. 1
    x1=0.5(10)+3=8x_1 = 0.5(10)+3 = 8; x2=0.5(8)+3=7x_2 = 0.5(8)+3 = 7; x3=0.5(7)+3=6.5x_3 = 0.5(7)+3 = 6.5.
  2. 2
    The sequence appears to converge. Fixed point: solve x∗=0.5x∗+3⇒0.5x∗=3⇒x∗=6x^* = 0.5x^*+3 \Rightarrow 0.5x^*=3 \Rightarrow x^*=6.
  3. 3
    Since ∣f′(x)∣=∣0.5∣<1|f'(x)|=|0.5|<1 at the fixed point, x∗=6x^*=6 is stable. The sequence converges to 66.

Answer

x1=8,x2=7,x3=6.5x_1=8, x_2=7, x_3=6.5; long-run value x∗=6x^*=6
A feedback map repeatedly applies a function to its previous output. If the map has a stable fixed point, iteration converges there regardless of starting value (within the basin of attraction).

About Feedback

Feedback occurs when the output of a system influences its future input — positive feedback amplifies changes; negative feedback stabilizes them.

Learn more about Feedback →

More Feedback Examples

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