###### GS 2003/Math - Collins
Lab 3
Patterns from Difference Equations

```In this lab we will use simulation to study various patterns
in sequences produced by difference equations of the form

x(n+1) = f(x(n))

We will look at different types of DE: Linear and Non-linear.

For each type of DE you will:
1. Write a script or function that accepts parameters and an
initial value, produces 100 iterates of the DE and
then plots the result.
2. Run your program many times with different parameters to
explore the behavior of the difference equation.
3. Formulate a conclusion saying what ranges of values for
the parameters and initial values produce what types of results.

I. Constant Linear DE: x(n+1) = ax(n) or x(n+1) = ax(n) + b

Program Info

The input to your script or function will be
a, b, and x(1)    [Matlab can only start at 1, not 0]

After you produce 100 iterates, plot the results as a series
of points (n,x(n)) for n = 1 to 100  (use plot(x,'o')
or plot(1:100,x,'o'))

Look back at the script from Lab 2 to see how your script might
look.  If you are still struggling, there is a script lindiff.m
at the end of this file.

Assignment I

By experimenting with a, b and x(1) determine for what parameter
ranges you get results that (a) stay bounded or (b) grow out
of control (unbounded).  When it is bounded, does it tend
toward one value?  What value?

II. Non-constant Linear DE: x(n+1) = a(n)x(n) + b(n)

Program Info

Several types of DEs in this category are listed below, for
each write a script to explore it.  Your script or function
should perform as described above. (You could/should copy the
program you wrote above and modify it as needed).

Assignment II

For each DE below, experiment with the parameters to see when
the sequence is (a) bounded and when it is (b) unbounded.  When
it is bounded, what value does it tend toward?

1. x(n+1) = a(1-1/n)x(n)
2. x(n+1) = a(1+1/n)x(n)
3. x(n+1) = a(1-1/n)x(n) + b
4. x(n+1) = a(1-1/n)x(n) + n

III. Nonlinear DE: x(n+1) = f(x(n))

Program Info

Same procedure as above.

Assignment III

For each non-linear DE below, experiment with the parameter to see
when the sequence is (a) bounded and when it is (b) unbounded.
When it is bounded, does the sequence tend towards one value?

1. x(n+1) = x(n)^2 + a,  use x(1) = 0
2. x(n+1) = a x(n) (1-x(n)), use 0 < a < 4, x(1) = 1/2
3. (Challenge) x(n+1) = a cos(x(n)), use x(1) = 0
4. (Challenge) x(n+1) = a x(n)    if 0 < x(n) < 1/2
a(1-x(n)) if 1/2 < x(n) < 1
Try a=2 then try other a < 2

IV. (Challenge) Complex-Valued Nonlinear DE: x(n+1) = f(x(n))

Program Info

Same as above except now the starting value and the parameters can
be complex numbers.  MATLAB naturally does complex arithmetic so
you don't need to change anything in your program.  To input
complex numbers you write them as 1+4i, 0.3-0.2i, etc.  When
MATLAB plots a complex number a+bi it plots it as the point (a,b),
thus to see the behavior you should use plot(x) which
will draw lines between the points.  'Bounded' in this case
means 'stays near the origin (0,0)', 'unbounded' means 'tends away from the
origin'.

Assignment IV (Challenge)

Classify the parameter ranges for bounded/unbounded for each:

1. x(n+1) = x(n)^2 + a, use x(1) = 0, but make  a  complex
2. x(n+1) = x(n) - abs(x(n))
3. x(n+1) = a cos(x(n))

To Turn In:

For each assignement/problem write out what parameter
ranges you found for each type of result: bounded,
unbounded, etc.

Sample Script

lindiff.m

% script file to simulate the difference equation x(n+1) = a x(n) + b

a = input('Enter the value for a:');
b = input('Enter the value for b:');
x(1) = input('Enter the initial value (x(1)):');

for n = 1:100
x(n+1) = a*x(n) + b;
end

plot(x,'o')
title(['Difference Equation with a = ',num2str(a),' and  b = ',num2str(b)])

% end of script

```
ccollins@math.utk.edu