Feedback Math Example 2

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Example 2

hard
Analyze the logistic map xn+1=3.5xn(1−xn)x_{n+1} = 3.5 x_n(1-x_n) by iterating from x0=0.5x_0=0.5 for five steps and commenting on the behavior.

Solution

  1. 1
    x1=3.5(0.5)(0.5)=0.875x_1 = 3.5(0.5)(0.5) = 0.875.
  2. 2
    x2=3.5(0.875)(0.125)=0.3828x_2 = 3.5(0.875)(0.125) = 0.3828; x3=3.5(0.3828)(0.6172)≈0.8269x_3 = 3.5(0.3828)(0.6172)\approx0.8269; x4≈3.5(0.8269)(0.1731)≈0.5009x_4\approx3.5(0.8269)(0.1731)\approx0.5009; x5≈0.8749x_5\approx0.8749.
  3. 3
    The sequence oscillates: 0.5,0.875,0.383,0.827,0.501,0.875,…0.5, 0.875, 0.383, 0.827, 0.501, 0.875, \ldots It appears to cycle between approximately 0.50.5 and 0.8750.875, exhibiting period-2 behavior typical of r=3.5r=3.5.

Answer

Iterates: 0.5,0.875,0.383,0.827,0.501,…0.5, 0.875, 0.383, 0.827, 0.501, \ldots; period-2 cycle
The logistic map with parameter r=3.5r=3.5 is in the period-doubling regime. Rather than converging to a fixed point, the iterates cycle between two values. This is a precursor to chaos in dynamical systems.

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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More Feedback Examples

Example 1 medium

Iterate the map [formula] starting from [formula]. Compute [formula] and predict the long-run value.

Example 3 easy

Iterate [formula] starting from [formula]. Compute [formula] and determine the long-run behavior.

Example 4 medium

Newton's method for finding [formula] uses [formula]. Starting from [formula], compute [formula] and

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