Math 5110/6830
Instructor: Alla Borisyuk
Homework 4.1
Due: September 22
1. Second-iterate equation for the logistic map
xn+1
= 2( n )
f
x
has right hand side
f
2
( )=
2
x
a) Show that = 0,
x
r
r
(1 ( + 1) + 2
r x
1
,
r+1
r
x
p
r
2
rx
(
3)(r +1)
2r
3
rx
)
:
are the xed points of this map.
b) Which of the xed points form the 2-cycle of the original system? For
what values of does it exist? What do the other points correspond to in the
original logistic equation?
c) Compute dxd 2 ( ).
p
d) Verify that thep non-trivial 2-cycle is stable for 3
1 + 6, and
unstable for
1 + 6.
r
f
x
< r <
r >
2. (MATLAB) a) For the logistic map n+1 = n (1 n ) numerically
nd 2 3
= 3 2 and 1 = 0 513. Plot n vs. . Did you nd
10 for
the 2-cycle? Also do the cobwebbing (plot functions in Matlab, then cobweb by
hand or in Matlab) to see the same 2-cycle.
b) Repeat a) with = 3 55 and 1 = 0 8874. What do you nd?
c) Repeat a) with = 3 8 and 1 = 0 5. What do you nd?
x
x ;x ;:::;x
r
:
r
:
r
:
x
:
x
rx
x
x
n
:
x
:
3. (MATLAB. Extra credit for 5110 students. Required for 6830
students) Generate Feigenbaum diagram of the logistic map. To do this, x
value of (say, = 0 1), then start with a hundred of dierent initial conditions
(100 of 1 values from 0 to 1), iterate each initial condition 50 times, and plot
50 as a function of . Repeat for dierent values of , between 0 and 4 to
see the stable part of the bifurcation diagram, with stable xed points, stable
2-cycles, 4-cycles and chaos. Label them in your graph (by hand is ne).
r
r
:
x
x
r
r
1