Math 5110/6830
Instructor: Alla Borisyuk
Homework 4.2
Due: September 26
1. Consider the Love Aairs model:
R
n+1
=
a
=
a
R Rn
+
R Jn ;
p
+ J n
Using initial conditions 0 = 0 = 1, plot in Matlab and print out the time
courses of the solutions and also show the same solutions in the phase plane for
the following paramater sets:
a) R = 0 5 J = 0 7 R = 0 2 J = 0 5.
b) R = 0 5 J = 0 7 R = 0 7 J = 0 9.
c) R = 1 J = 1 R = 0 2 J = 0 2.
d) R = 0 5 J = 0 8 R = 0 2 J = 0 5.
2. For the discrete logistic model
n+1
J
R
: ;a
: ;p
: ;p
a
: ;a
: ;p
: ;p
a
;a
a
: ;a
:
:
: ;p
: ;p
:
: ;p
n+1
x
a)Plot rst 4 iterations (i.e.
p R :
J
a
;p
J Jn
:
= (1
r
x
n
K
)
x
n
:
up to 4 ) of a cobwebbing diagram for = 2 8 =
1 0 = 0 3.
b)Explain just by looking at your diagram from a) the stability of all xed
points.
c)Plot the solution (up to time 4) that you found in a) as a function of time in
a separate gure.
d)Find (from the notes is ne) the stability of the xed points analytically and
make a plot of the bifurcation diagram. (You do need to include cycles in this
diagram)
e)(extra credit) Plot the orbital bifurcation diagram (Feigenbaum diagram)
for the discrete logistic equation. Describe what procedure you use.
;x
x
r
:
1
: ;K