Stability Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

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.

Solution

1
Fixed point: $x^* = \frac{1}{2}x^*+4 \Rightarrow \frac{1}{2}x^*=4 \Rightarrow x^*=8$ . $|h'(x)|=|\frac{1}{2}|=0.5<1$ : stable.
2
From $x_0=0$ : $x_1=4, x_2=6, x_3=7$ . From $x_0=20$ : $x_1=14, x_2=11, x_3=9.5$ . Both sequences approach $8$ .

Answer

Fixed point

x^*=8

(stable); both orbits converge toward

8

A stable fixed point acts as an attractor. Regardless of starting point, iteration of a contraction map converges to the unique fixed point. The convergence rate is governed by

|f'(x^*)|=0.5

per step.

About Stability

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.

Learn more about Stability →

More Stability Examples

Example 1 medium

Find all fixed points of [formula] and determine their stability using the derivative criterion [for

Example 2 hard

For the map [formula], find the fixed point (Dottie number) approximately and determine its stabilit

Example 3 easy

Classify the fixed points of [formula] as stable or unstable using the derivative criterion.

← All Stability Examples Stability Concept

Related Concepts

Function