Feedback Math Example 3

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

easy

Iterate

x_{n+1} = x_n^2

starting from

x_0 = 0.5

. Compute

x_1, x_2, x_3

and determine the long-run behavior.

1
$x_1 = (0.5)^2 = 0.25$ ; $x_2 = (0.25)^2 = 0.0625$ ; $x_3 = (0.0625)^2 \approx 0.0039$ .
2
The iterates are rapidly approaching $0$ . Since $|x_0|=0.5<1$ , repeated squaring sends the sequence to $0$ .

Answer

x_1=0.25, x_2=0.0625, x_3\approx0.0039

; converges to

0

For

x_{n+1}=x_n^2

, if

|x_0|<1

the iterates decay to

0

; if

|x_0|>1

they diverge to

\infty

. The fixed points are

0

(stable) and

1

(unstable).

About Feedback

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

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