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.
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)=x2−x+1 and determine their stability using the derivative criterion ∣f′(x∗)∣<1.
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 vn+1=vn+0.4(60−vn). Identify equilibrium and classify stability.
Example 5
hard
Find and classify the equilibrium of xn+1=xn−0.1(xn−3)3 near x∗=3.
Example 6
challenge
A controlled inverted pendulum obeys xn+1=2xn−k⋅xn=(2−k)xn 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)=21x+4, find the fixed point and verify stability by iterating from x0=0 and x0=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 xn+1=0.5xn has equilibrium x=0. Starting at x0=8, compute x1,x2,x3. Is x=0 stable?
Example 7
easy
The recurrence xn+1=2xn has equilibrium x=0. Starting at x0=1, compute x1,x2,x3. 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 xn+1=rxn 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 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 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 Bn+1=1.05Bn−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 xn+1=rxn(1−xn) 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 xn+1=xn+c(5−xn). 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: xn+1=0.5xn+0.5yn, yn+1=0.5xn+0.5yn. Show the total xn+yn is conserved and determine whether the equal-split state is stable.
Example 21
medium
For xn+1=rxn 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 xn+1=0.3xn with equilibrium x=0, is x=0 stable?
Example 24
easy
For the map xn+1=1.4xn, is the fixed point x=0 stable?
Example 25
easy
Find the fixed point of f(x)=21x+3.
Example 26
easy
True or false: a stable equilibrium means the system never moves.
Example 27
medium
For the recurrence xn+1=−0.5xn with x∗=0, classify stability.
Example 28
medium
For xn+1=−1.2xn with x∗=0, classify stability.
Example 29
medium
For f(x)=x2, classify the fixed point x∗=0.
Example 30
medium
Find all fixed points of f(x)=x2−2x+2 and classify each.
Example 31
medium
A drug-clearance model is Cn+1=0.7Cn+50 (residual after dosing). Find the equilibrium and decide stability.
Example 32
hard
For the Newton iteration xn+1=xn−f′(xn)f(xn) with f(x)=x2−2, classify the fixed point x∗=2.
Example 33
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 34
hard
A savings account uses Bn+1=1.02Bn−W (2% growth minus withdrawal W). Find the equilibrium and classify stability.
Example 35
challenge
For the iteration xn+1=xn2+c with c=−1, find the fixed points and classify each.