Practice Stability in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium
For the recurrence xn+1=−0.5xn with x∗=0, classify stability.

Example 2

medium
A logistic-type map has a fixed point at x∗ with local multiplier (slope of the update at x∗) equal to −0.4. Is x∗ locally stable, and roughly how does the approach look?

Example 3

medium
A ball rolls in a double-well: two valleys separated by a hill. List which of the three special points are stable and which is unstable.

Example 4

medium
For f(x)=x2, classify the fixed point x∗=1.

Example 5

easy
Which sign of feedback typically produces a stable equilibrium?

Example 6

hard
A predator-prey simplification gives xn+1=xn+h(αxn−βxnyn). Linearizing about (x∗,y∗) with eigenvalues 1±iθ (θ small) signals what kind of behavior?

Example 7

easy
True or false: a stable equilibrium means the system never moves.

Example 8

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

Example 9

challenge
For the iteration xn+1=xn2+c with c=−1, find the fixed points and classify each.

Example 10

hard
Find and classify the equilibrium of xn+1=xn−0.1(xn−3)3 near x∗=3.

Example 11

easy
The recurrence xn+1=0.5xn has equilibrium x=0. Starting at x0=8, compute x1,x2,x3. Is x=0 stable?

Example 12

easy
Find the fixed point of f(x)=12x+3.

Example 13

medium
For xn+1=−1.2xn with x∗=0, classify stability.

Example 14

hard
For the Newton iteration xn+1=xn−f(xn)f′(xn) with f(x)=x2−2, classify the fixed point x∗=2.

Example 15

easy
A marble sits on a perfectly flat tabletop. Tapped, it rolls to a new spot. Is this stable, unstable, or neutral?

Example 16

easy
The recurrence xn+1=2xn has equilibrium x=0. Starting at x0=1, compute x1,x2,x3. Is x=0 stable?

Example 17

hard
For the logistic map f(x)=rx(1−x), the nonzero fixed point x∗=1−1/r has multiplier 2−r. For what r is this fixed point exactly at the stability boundary?

Example 18

easy
A ball sits in a hollow bowl. Tap it lightly. Is the bottom of the bowl a stable equilibrium?

Example 19

medium
The same map xn+1=xn2 has fixed points at x=0 and x=1. Starting at x0=0.5, where does it go, and is x=0 stable?

Example 20

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