Applied Dynamical Systems - ME215A Fall 2010
Take-Home Final - Part I - Due Wednesday December 1
1. (45 pts total) Consider the vector eld
x = (x + 1)(x + 2y )(1 (x + y 1)/5),
y = (y + 1)(2x + y ).
(1)
(2)
(a) (5 pts) Verify that there are xed points at (
Applied Dynamical Systems - ME215A Fall 2010
Take-Home Final Part II - due Wednesday December 8
1. (10 pts) Write a Matlab program to generate a classic bifurcation diagram for the
logistic map
xn+1 = xn (1 xn ).
(1)
for 2.8 < < 4.
2. (25 pts total) Consi
Applied Dynamical Systems - ME215A
Fall 2010
Homework #1 - Due Wednesday Oct 6th, in class
1. (20 pts total) Consider the linear set of ordinary dierential equations for x = (x, y ):
x = Ax,
(1)
where
A=
1 100
0 10
.
(2)
(a) (2 pts) Show that A is a non-n
Applied Dynamical Systems - ME215A
Fall 2010
Homework #2 - Due Wednesday Oct 13th in class
1. (5 pts total) Suppose
M=
AB
0D
.
(1)
Verify that the eigenvalues of M are A and D, with associated eigenvectors (1, 0)T and
(B/(D A), 1)T , respectively.
2. (45
Applied Dynamical Systems - ME215A
Fall 2010
Homework #3 - Due Wednesday October 20, in class
1. (15 pts total) Consider the map
xn+1 = xn (1 xn ) g (xn ).
(1)
(a) (5 pts) Find the xed points of this map. Your expressions may depend on .
(b) (5 pts) Over
Applied Dynamical Systems - ME215A
Fall 2010
Homework #4 - Due Wednesday October 27, in class
1. (20 pts total) Consider the vector eld
x = x x2 ,
y = y.
(1)
(2)
(a) (3 pts) What is the -limit set and -limit set for this ow?
(b) (3 pts) What is the nonwan
Applied Dynamical Systems - ME215A Fall 2010
Homework #5 - Due Wednesday November 3rd in class
1. (20 pts total) Consider the system
x = x + y x3 6xy 2
x
y = + 2y 8y 3 x2 y
2
(1)
(2)
(a) (5 pts) Show that (x, y ) = (0, 0) is an unstable xed point.
(b) (10
Applied Dynamical Systems - ME215A
Fall 2010
Homework 6 - Due Wednesday November 10 - in class
1. (20 pts total) Consider the vector eld
x = y 3x2 + xy,
y = 3y + y 2 + x2 .
(1)
(2)
(a) (5 pts) What are the eigenvalues of the xed point at the origin? Find
Applied Dynamical Systems - ME215A
Fall 2010
Homework 7 - Due Wednesday November 17
1. (25 pts) Consider the vector eld
x = y + a1 x2 + b1 xy + c1 y 2 ,
y = a2 x2 + b2 xy + c2 y 2 ,
(1)
(2)
whose Jacobian evaluated at the xed point (x, y ) = (0, 0) is
01