Stability Math Example 1

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Example 1

medium

Find all fixed points of

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

and determine their stability using the derivative criterion

|f'(x^*)| < 1

Solution

1
Fixed points: solve $x = x^2-x+1 \Rightarrow x^2-2x+1=0 \Rightarrow (x-1)^2=0 \Rightarrow x^*=1$ (double root).
2
Compute $f'(x)=2x-1$ . At $x^*=1$ : $|f'(1)|=|2(1)-1|=|1|=1$ .
3
Since $|f'(x^*)|=1$ , the stability criterion is inconclusive (marginal stability). Numerical experimentation would be needed to determine behavior.

Answer

Fixed point

x^*=1

;

|f'(1)|=1

— marginal stability

The stability of a fixed point depends on

|f'(x^*)|

<1

means stable (attracting),

>1

means unstable (repelling),

=1

is inconclusive. A double root gives

|f'|=1

, a degenerate case.

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 2 hard

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

Example 3 easy

Classify the fixed points of [formula] as stable or unstable using the derivative criterion.

Example 4 medium

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

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Function