Stability Formula

Stability is a system is stable at an equilibrium if small perturbations cause it to return toward that equilibrium.

The Formula

f(x∗)=x∗f(x^*) = x^* (equilibrium) with ∣f′(x∗)∣<1|f'(x^*)| < 1 (stable) or ∣f′(x∗)∣>1|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∗x^* denotes an equilibrium point where f(x∗)=x∗f(x^*) = x^*. Stability is determined by ∣f′(x∗)∣|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∗x^* is a stable fixed point of ff   ⟺  \iff f(x∗)=x∗f(x^*) = x^* and ∣f′(x∗)∣<1|f'(x^*)| < 1; unstable if ∣f′(x∗)∣>1|f'(x^*)| > 1

Worked Examples

Example 1

medium
Find all fixed points of f(x)=x2−x+1f(x) = x^2 - x + 1 and determine their stability using the derivative criterion ∣f′(x∗)∣<1|f'(x^*)| < 1.

Answer

Fixed point x∗=1x^*=1; ∣f′(1)∣=1|f'(1)|=1 — marginal stability

First step

1
Fixed points: solve x=x2−x+1⇒x2−2x+1=0⇒(x−1)2=0⇒x∗=1x = x^2-x+1 \Rightarrow x^2-2x+1=0 \Rightarrow (x-1)^2=0 \Rightarrow x^*=1 (double root).

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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.

Example 3

medium
Find the fixed point of f(x)=2x−6f(x)=2x-6 and determine its stability.

Common Mistakes

  • Confusing where the equilibrium is with whether it's stable - first solve f(x∗)=x∗f(x^*)=x^*, then test the slope.
  • Using f(x∗)f(x^*) instead of f′(x∗)f'(x^*) for the test - stability depends on the derivative's magnitude, not the function value.
  • Forgetting the absolute value - ∣f′(x∗)∣<1|f'(x^*)|<1 is stable even if f′(x∗)f'(x^*) is negative (which adds oscillation).

Why This Formula Matters

Stability is the payoff question for any feedback system, recurrence, or equilibrium: will it stay put or collapse? The slope test ∣f′(x∗)∣<1|f'(x^*)|<1 turns a vague 'does it settle?' into a checkable condition, central to dynamics, economics, and ecology. Recognizing it by "After a small nudge, does the system move back toward the equilibrium rather than away from it?" — rather than by familiar numbers — is what lets a student tell it apart from equilibrium / fixed point and feedback and convergence of a sequence in a mixed problem set.

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

Why is the Stability formula important in Math?

Stability is the payoff question for any feedback system, recurrence, or equilibrium: will it stay put or collapse? The slope test ∣f′(x∗)∣<1|f'(x^*)|<1 turns a vague 'does it settle?' into a checkable condition, central to dynamics, economics, and ecology. Recognizing it by "After a small nudge, does the system move back toward the equilibrium rather than away from it?" — rather than by familiar numbers — is what lets a student tell it apart from equilibrium / fixed point and feedback and convergence of a sequence in a mixed problem set.

What do students get wrong about Stability?

The procedure for stability is the easy part; the trap is confusing where the equilibrium is with whether it's stable. Asking "After a small nudge, does the system move back toward the equilibrium rather than away from it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Stability formula?

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

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Function

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