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 y = f (x) and the line y = x 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: Use cobwebbing to determine the limiting behavior of
xt+1 = 2xt (1 − xt )
for the given initial condition.
(a) x0 = 0.25
(b) x0 = −0.25
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Example: Consider the difference equation
1
xt+1 = xt + 1.
2
(a) Find the equilibrium, x∗ , of this difference equation.
(b) Use cobwebbing to determine the stability of x∗ if x0 = 4. What if x0 = 0.5?
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Example: Consider the difference equation for exponential growth in discrete time
Nt+1 = RNt ,
where R > 0 is the growth rate. If R 6= 1, the only equilibrium is N ∗ = 0.
(a) Determine the stability of N ∗ = 0 for R > 1.
(b) Determine the stability of N ∗ = 0 for 0 < R < 1.
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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.
Example: Find all equilibria of the difference equation
3
2
xt+1 = x2t −
5
5
and discuss their stability.
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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) Characterize the stability of the equilibria of this difference equation.
(b) Use cobwebbing to decide which value of xt the difference equation converges to if
x0 = 1 and if x0 = 5.
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