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=โˆ’0.5xnx_{n+1}=-0.5 x_n with xโˆ—=0x^*=0, classify stability.

Example 2

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A logistic-type map has a fixed point at xโˆ—x^* with local multiplier (slope of the update at xโˆ—x^*) equal to โˆ’0.4-0.4. Is xโˆ—x^* locally stable, and roughly how does the approach look?

Example 3

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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)=x2f(x)=x^2, classify the fixed point xโˆ—=1x^*=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(ฮฑxnโˆ’ฮฒxnyn)x_{n+1}=x_n+h(\alpha x_n-\beta x_n y_n). Linearizing about (xโˆ—,yโˆ—)(x^*,y^*) with eigenvalues 1ยฑiฮธ1\pm i\theta (ฮธ\theta 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)=cosโก(x)g(x) = \cos(x), find the fixed point (Dottie number) approximately and determine its stability.

Example 9

challenge
For the iteration xn+1=xn2+cx_{n+1}=x_n^2+c with c=โˆ’1c=-1, find the fixed points and classify each.

Example 10

hard
Find and classify the equilibrium of xn+1=xnโˆ’0.1(xnโˆ’3)3x_{n+1}=x_n-0.1(x_n-3)^3 near xโˆ—=3x^*=3.

Example 11

easy
The recurrence xn+1=0.5xnx_{n+1} = 0.5 x_n has equilibrium x=0x = 0. Starting at x0=8x_0 = 8, compute x1,x2,x3x_1, x_2, x_3. Is x=0x=0 stable?

Example 12

easy
Find the fixed point of f(x)=12x+3f(x)=\tfrac{1}{2}x+3.

Example 13

medium
For xn+1=โˆ’1.2xnx_{n+1}=-1.2 x_n with xโˆ—=0x^*=0, classify stability.

Example 14

hard
For the Newton iteration xn+1=xnโˆ’f(xn)fโ€ฒ(xn)x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)} with f(x)=x2โˆ’2f(x)=x^2-2, classify the fixed point xโˆ—=2x^*=\sqrt 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=2xnx_{n+1} = 2 x_n has equilibrium x=0x = 0. Starting at x0=1x_0 = 1, compute x1,x2,x3x_1, x_2, x_3. Is x=0x=0 stable?

Example 17

hard
For the logistic map f(x)=rx(1โˆ’x)f(x)=rx(1-x), the nonzero fixed point xโˆ—=1โˆ’1/rx^*=1-1/r has multiplier 2โˆ’r2-r. For what rr 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=xn2x_{n+1} = x_n^2 has fixed points at x=0x = 0 and x=1x = 1. Starting at x0=0.5x_0 = 0.5, where does it go, and is x=0x=0 stable?

Example 20

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Find the fixed point of f(x)=2xโˆ’6f(x)=2x-6 and determine its stability.
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Related Concepts

Function