Stability Math Example 2

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Example 2

hard
For the map g(x)=cos⁡(x)g(x) = \cos(x), find the fixed point (Dottie number) approximately and determine its stability.

Solution

  1. 1
    Fixed point: solve cos⁡(x∗)=x∗\cos(x^*) = x^*. There is no closed-form solution; numerically, x∗≈0.7391x^* \approx 0.7391 radians (the Dottie number).
  2. 2
    Stability: g′(x)=−sin⁡(x)g'(x) = -\sin(x). At x∗≈0.7391x^*\approx0.7391: ∣g′(0.7391)∣=∣−sin⁡(0.7391)∣=sin⁡(0.7391)≈0.6736<1|g'(0.7391)|=|{-}\sin(0.7391)|=\sin(0.7391)\approx0.6736<1.
  3. 3
    Since ∣g′(x∗)∣≈0.6736<1|g'(x^*)|\approx0.6736<1, the fixed point is stable. Iterating cos⁡\cos from any starting angle (in radians) converges to ≈0.7391\approx0.7391.

Answer

Fixed point x∗≈0.7391x^*\approx0.7391; stable since ∣g′(x∗)∣≈0.674<1|g'(x^*)|\approx0.674<1
The Dottie number is the unique real fixed point of cosine. Its stability explains why repeatedly pressing 'cos' on a calculator (in radian mode) always converges to approximately 0.73910.7391.

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 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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