Page 1 | © 2014 by Janice L. Epstein 5.6 Difference Equations: Stability 5.6: Difference Equations: Stability
In Section 2.2 we studies recursions that looked like Nt 1  f  Nt  .
Written in the form Nt 1  Nt  g  Nt  they are difference equations.
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Example: Is the fixed point a  2 the limit for the recursion an 1  an  1
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Look at this graphically using cobwebbing which is a technique for
determining the limiting behavior of a sequence based on different initial
conditions (that is, different values of a0). Graph below shows an+1 as a
function of an. Note the intersection is the fixed point,  an , an1    2, 2 
How to cobweb:
1. Graph the recursion equation and g  a   a . Plot the point  a0 , a0 
which will be on the line g  a   a .
2. Draw a vertical line from the point  a0 , a0  to the point  a0 , a1  which
will be on the recursion line.
3. Draw a horizontal line from  a0 , a1  to the line g  a   a which will
be the point  a1 , a1  .
4. Continue until the limiting behavior can be determined.
Page 2 | © 2014 by Janice L. Epstein 5.6 Difference Equations: Stability The equilibrium a  2 is asymptotically (locally) stable since a choice of
a0 near a  2 results in a sequence that approaches and then remains at
a  2.
Example:
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The recursion an 1  2an 1  an  has fixed points at a  0 and a  .
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(a) Use the method of cobwebbing to find lim an if a0 
n¥
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(b) Repeat for a0 
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and a0  
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Page 3 | © 2014 by Janice L. Epstein 5.6 Difference Equations: Stability Example: Use cobwebbing to discuss the stability of the equilibrium a  0
for the recursion an 1  Ran for the following cases:
(a) 0  R  1
(b) R  1
(c) 1  R  0
(e) R  1
(d) R  1
Page 4 | © 2014 by Janice L. Epstein 5.6 Difference Equations: Stability Is there a way to determine stability without cobwebbing? Yes, but we
will only look “near” the fixed points to see if they are stable or unstable.
Suppose a sequence is defined recursively as xn 1  f  xn  and x* is a
fixed point. Our starting point will be xt  x*  zt . The term zt represents
a small perturbation from the fixed point.
Since we are looking near the fixed point, the function will be
approximately linear.
This result has the same form as an 1  Ran , so we can use the result of the
previous page to decide if a fixed point is stable or not.
Stability Criterion for Equilibria
An equilibrium x* of xn 1  f  xn  is asymptotically stable if f   x*   1
Page 5 | © 2014 by Janice L. Epstein 5.6 Difference Equations: Stability Example: Use the stability criterion to characterize the stability of the
equilibria of
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(a) xt 1  2 xt 1  xt  , t  0,1, 2, and (b) xt 1  xt 2  , t  0,1, 2,
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