Nonlinear, recursive difference equation

Nonlinear difference equations can behave very unpredictably, and are much harder to analyze than linear difference equation. Below we illustrate this with a simple example of the following difference equation, where [Graphics:../Images/index_gr_108.gif] is some real number greater than 0.

[Graphics:../Images/index_gr_109.gif]

Experiment #1

Difference equation definition 
[Graphics:../Images/index_gr_110.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_111.gif]
Simulation result 
[Graphics:../Images/index_gr_112.gif]

[Graphics:../Images/index_gr_113.gif]

Experiment #2

Difference equation definition 
[Graphics:../Images/index_gr_114.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_115.gif]
Simulation result 
[Graphics:../Images/index_gr_116.gif]

[Graphics:../Images/index_gr_117.gif]

Experiment #3

Difference equation definition 
[Graphics:../Images/index_gr_118.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_119.gif]
Simulation result 
[Graphics:../Images/index_gr_120.gif]

[Graphics:../Images/index_gr_121.gif]

Experiment #4

Difference equation definition 
[Graphics:../Images/index_gr_122.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_123.gif]
Simulation result 
[Graphics:../Images/index_gr_124.gif]

[Graphics:../Images/index_gr_125.gif]

Experiment #5

Difference equation definition 
[Graphics:../Images/index_gr_126.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_127.gif]
Simulation result 
[Graphics:../Images/index_gr_128.gif]

[Graphics:../Images/index_gr_129.gif]

Experiment #6

Difference equation definition 
[Graphics:../Images/index_gr_130.gif]
Initial condition and constant R 
[Graphics:../Images/index_gr_131.gif]
Simulation result 
[Graphics:../Images/index_gr_132.gif]

[Graphics:../Images/index_gr_133.gif]

All results

[Graphics:../Images/index_gr_134.gif] 

[Graphics:../Images/index_gr_135.gif]

[Graphics:../Images/index_gr_136.gif]

Converted by Mathematica      January 28, 2003