Stability Math Example 3

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

Example 3

easy

Classify the fixed points of

f(x) = 2x(1-x)

as stable or unstable using the derivative criterion.

Solution

1
Fixed points: $x = 2x(1-x) \Rightarrow x=2x-2x^2 \Rightarrow 2x^2-x=0 \Rightarrow x(2x-1)=0 \Rightarrow x^*=0$ or $x^*=\frac{1}{2}$ .
2
$f'(x)=2-4x$ . At $x^*=0$ : $|f'(0)|=2>1$ (unstable). At $x^*=\frac{1}{2}$ : $|f'(\frac{1}{2})|=|2-2|=0<1$ (stable).

Answer

x^*=0

is unstable;

x^*=\frac{1}{2}

is stable

Fixed points with

|f'(x^*)|<1

attract nearby iterates; those with

|f'(x^*)|>1

repel them. Here the origin repels while

x^*=1/2

attracts: orbits starting near

1/2

converge there.

About Stability

A system is stable at an equilibrium if small perturbations cause it to return toward that equilibrium; unstable if small perturbations cause it to move away.

Learn more about Stability →

More Stability Examples

Example 1 medium

Find all fixed points of [formula] and determine their stability using the derivative criterion [for

Example 2 hard

For the map [formula], find the fixed point (Dottie number) approximately and determine its stabilit

Example 4 medium

For the map [formula], find the fixed point and verify stability by iterating from [formula] and [fo

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