Section 5.6: Stability of Difference Equations
The general form of a first-order difference equation is
xt+1 = f (xt )
Definition: An equilibrium or fixed point of a difference equation xt+1 = f (xt ) is a value
x∗ that is unchanged by the function f . That is,
x∗ = f (x∗ )
An equilibrium is called locally stable if solutions that begin close to the equilibrium
approach that equilibrium and unstable if solutions that start close to the equilibrium
move away from it.
Example: Find the equilibria of xt+1 = 2xt (1 − xt ) and determine the limiting value of xt
given that x0 = 0.25.
There is a graphical method for finding equilibria and determining their stability.
1. Graph the recursion equation xt+1 = f (xt ) and the line xt+1 = xt together.
2. Start at x0 on the horizontal axis and find the corresponding value x1 = f (x0 ) on the
recursion graph.
3. Find x1 on the horizontal axis and find the corresponding value x2 = f (x1 ) on the
recursion graph.
4. Repeat this process until the limiting behavior can be determined.
This method is known as cobwebbing.
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Example: Consider the difference equation
xt+1 = 2xt (1 − xt ),
x0 = 0.25
Find the equilibria and use cobwebbing to determine their stability.
Example: Consider the difference equation
xt+1 = 0.5xt + 1,
x0 = 4
Find the equilibria and use cobwebbing to determine their stability.
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Example: Find the fixed point of each difference equation and use cobwebbing to determine
its stability.
(a) xt+1 = 3 − 2xt
(b) xt+1 = 1 − 0.5xt
(c) xt+1 =
1
xt
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Theorem: (Stability Criterion)
An equilibrium x∗ of the difference equation xt+1 = f (xt ) is locally stable if
|f 0 (x∗ )| < 1
Moreover, if f 0 (x∗ ) > 0, then x∗ is approached without oscillations and if f 0 (x∗ ) < 0, then
x∗ is approached with oscillations.
Oscillations
No Oscillations
35
30
30
25
25
20
20
xt
xt
35
15
15
10
10
5
5
0
0
0
5
10
15
20
0
t
5
10
t
Example: Find all equilibria of the difference equation
3
2
xt+1 = x2t −
5
5
and discuss their stability.
4
15
20
Example: Find all equilibria of the difference equation
xt+1 =
xt
0.3 + xt
and discuss their stability.
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Example: Consider the difference equation
xt+1
11x2t
=
18 + x2t
(a) Find all equilibria and discuss their stability.
(b) Use cobwebbing to find lim xt if x0 = 1 or if x0 = 5.
t→∞
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