Stability Formula

The Formula

f(x^*) = x^* (equilibrium) with |f'(x^*)| < 1 (stable) or |f'(x^*)| > 1 (unstable)

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

Quick Example

Pendulum at rest is stable (returns after push). Balanced pencil is unstable.

Notation

x^* denotes an equilibrium point where f(x^*) = x^*. Stability is determined by |f'(x^*)|.

What This Formula Means

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.

Formal View

x^* is a stable fixed point of f \iff f(x^*) = x^* and |f'(x^*)| < 1; unstable if |f'(x^*)| > 1

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.

Common Mistakes

  • Thinking stable means unchanging — a stable equilibrium can be reached through oscillation; stability means returning to equilibrium after perturbation
  • Confusing local stability with global stability — a system can be locally stable (small pushes return) but globally unstable (large pushes diverge)
  • Ignoring initial conditions — the same system can be stable or unstable depending on where it starts

Why This Formula Matters

Stability analysis determines whether a designed system (bridge, circuit, algorithm, ecosystem) will behave reliably under small disturbances or catastrophically diverge.

Frequently Asked Questions

What is the Stability formula?

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.

How do you use the Stability formula?

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

What do the symbols mean in the Stability formula?

x^* denotes an equilibrium point where f(x^*) = x^*. Stability is determined by |f'(x^*)|.

Why is the Stability formula important in Math?

Stability analysis determines whether a designed system (bridge, circuit, algorithm, ecosystem) will behave reliably under small disturbances or catastrophically diverge.

What do students get wrong about Stability?

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

What should I learn before the Stability formula?

Before studying the Stability formula, you should understand: function definition.

← Back to Stability See All Examples Practice Problems

Related Concepts

Function

Related Formulas

Function Formula