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.

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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: At an equilibrium, a stable system pulls small disturbances back toward the point; an unstable one pushes them away.

Common stuck point: 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.

Sense of Study hint: Ask: After a small nudge, does the system move back toward the equilibrium rather than away from it?

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

Answer

Fixed point

x^*=1

;

|f'(1)|=1

— marginal stability

First step

Fixed points: solve

x = 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)

, find the fixed point (Dottie number) approximately and determine its stability.

Example 3

medium

Find the fixed point of

f(x)=2x-6

and determine its stability.

Example 4

medium

A car's cruise control adjusts speed as

v_{n+1}=v_n+0.4(60-v_n)

. Identify equilibrium and classify stability.

Example 5

hard

Find and classify the equilibrium of

x_{n+1}=x_n-0.1(x_n-3)^3

near

x^*=3

Example 6

challenge

A controlled inverted pendulum obeys

x_{n+1}=2 x_n - k\cdot x_n=(2-k) x_n

after adding feedback strength

k

. For what range of

k

is the upright position stable?

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.

Example 3

easy

A ball rests at the bottom of a bowl. Nudged slightly, it rolls back to the center. Is this equilibrium stable or unstable?

Example 4

easy

A ball is balanced on the top of a smooth hill. A tiny nudge sends it rolling away and it never returns. Stable or unstable?

Example 5

easy

A pendulum hangs straight down at rest. Pushed slightly, it swings and eventually settles back to hanging down. Is the hanging-down position stable?

Example 6

easy

The recurrence

x_{n+1} = 0.5 x_n

has equilibrium

x = 0

. Starting at

x_0 = 8

, compute

x_1, x_2, x_3

. Is

x=0

stable?

Example 7

easy

The recurrence

x_{n+1} = 2 x_n

has equilibrium

x = 0

. Starting at

x_0 = 1

, compute

x_1, x_2, x_3

. Is

x=0

stable?

Example 8

easy

True or false: a stable equilibrium means the system can never move once it gets there.

Example 9

easy

A marble sits on a flat horizontal table. Nudged, it rolls to a new spot and stays there (ignoring friction stopping it). Is this equilibrium stable, unstable, or neutral?

Example 10

easy

Which sign of feedback typically produces a stable equilibrium?

Example 11

medium

For

x_{n+1} = r x_n

with equilibrium

x=0

, state the exact range of

r

for which the equilibrium is stable, and classify

r = -0.7

Example 12

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 13

medium

A system is locally stable near

x = 0

but a large enough push sends it diverging. The behavior 'returns if pushed by less than 3, diverges if pushed more' describes which combination?

Example 14

medium

The same map

x_{n+1} = x_n^2

has fixed points at

x = 0

and

x = 1

. Starting at

x_0 = 0.5

, where does it go, and is

x=0

stable?

Example 15

medium

An inverted pendulum (balanced straight up) is held by a controller that pushes the base toward the lean. Without the controller, is the upright position stable? With a strong correcting controller, can it become stable?

Example 16

medium

A savings model

B_{n+1} = 1.05 B_n - 100

(5% growth minus a $100 withdrawal). Find the equilibrium balance and decide whether it is stable.

Example 17

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 18

challenge

For the map

x_{n+1} = r x_n (1 - x_n)

the nonzero fixed point is

x^* = 1 - 1/r

. The local multiplier there is

2 - r

. For what range of

r

(with

r>1

) is this fixed point stable?

Example 19

challenge

A system has

x_{n+1} = x_n + c(5 - x_n)

. Determine the range of

c

for which the equilibrium

x=5

is stable, and identify the

c

giving fastest (one-step) convergence.

Example 20

challenge

Two coupled towns share population:

x_{n+1} = 0.5x_n + 0.5y_n

y_{n+1} = 0.5x_n + 0.5y_n

. Show the total

x_n + y_n

is conserved and determine whether the equal-split state is stable.

Example 21

medium

For

x_{n+1} = r x_n

with equilibrium 0, classify the stability for

r = 1

exactly (the boundary case).

Example 22

medium

A damped pendulum is pushed and swings with shrinking oscillations until it rests straight down. Is the rest position stable, and does 'stable' allow oscillation on the way back?

Example 23

easy

For the map

x_{n+1}=0.3 x_n

with equilibrium

x=0

, is

x=0

stable?

Example 24

easy

For the map

x_{n+1}=1.4 x_n

, is the fixed point

x=0

stable?

Example 25

easy

Find the fixed point of

f(x)=\tfrac{1}{2}x+3

Example 26

easy

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

Example 27

medium

For the recurrence

x_{n+1}=-0.5 x_n

with

x^*=0

, classify stability.

Example 28

medium

For

x_{n+1}=-1.2 x_n

with

x^*=0

, classify stability.

Example 29

medium

For

f(x)=x^2

, classify the fixed point

x^*=0

Example 30

medium

Find all fixed points of

f(x)=x^2-2x+2

and classify each.

Example 31

medium

A drug-clearance model is

C_{n+1}=0.7 C_n + 50

(residual after dosing). Find the equilibrium and decide stability.

Example 32

hard

For the Newton iteration

x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}

with

f(x)=x^2-2

, classify the fixed point

x^*=\sqrt 2

Example 33

hard

A predator-prey simplification gives

x_{n+1}=x_n+h(\alpha x_n-\beta x_n y_n)

. Linearizing about

(x^*,y^*)

with eigenvalues

1\pm i\theta

(

\theta

small) signals what kind of behavior?

Example 34

hard

A savings account uses

B_{n+1}=1.02 B_n - W

(2% growth minus withdrawal

W

). Find the equilibrium and classify stability.

Example 35

challenge

For the iteration

x_{n+1}=x_n^2+c

with

c=-1

, find the fixed points and classify each.

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Related Concepts

Function

Background Knowledge

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

function definition