Stability Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Stability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

A ball in a bowl returns to center; a ball on a hill rolls away.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Stability is determined by the sign of the feedback at the equilibrium — negative feedback (restoring force) gives stability; positive feedback (amplifying force) gives instability.

Common stuck point: An equilibrium being stable does not mean the system stays exactly there — it means small disturbances decay rather than grow.

Sense of Study hint: Imagine giving the system a small push. Does it return to where it was (stable) or drift further away (unstable)?

Worked Examples

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. 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. 2
    Compute f'(x)=2x-1. At x^*=1: |f'(1)|=|2(1)-1|=|1|=1.
  3. 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.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Classify the fixed points of f(x) = 2x(1-x) as stable or unstable using the derivative criterion.

Example 2

medium
For the map h(x) = \frac{1}{2}x + 4, find the fixed point and verify stability by iterating from x_0 = 0 and x_0 = 20 for three steps each.
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Related Concepts

Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition