MAT 431 - WINTER 2012
Project 1
Ryan Szypowski
Due February 12, 2013
Instructions
In this project, you will study a particular model of a shery. The mathematical details are
given in Strogatz question 3.7.3 and repeated below.
The deliverable for the proj
MAT 431 - WINTER 2013
Homework 4 Solutions
Ryan Szypowski
Due March 7, 2013
1. Prove that the system
x = x y x(x2 + y 2 )
y = x + y y(x2 + y 2 )
has a closed orbit for any > 0 using the Poincar-Bendixson theorem. It may be
e
easiest to convert this proble
MAT 431 - WINTER 2012
Project 2
Ryan Szypowski
Due March 14, 2013
Instructions
In this project, you will study a particular model of a chemical oscillator. The mathematical
details are given in Strogatz question 8.3.1 and repeated below.
The deliverable f
MAT 431 - WINTER 2013
Homework 3 Solutions
Ryan Szypowski
Due Feb 26, 2013
1. Let f : R2 R be dened by
f (x, y) = x4 2x2 + y 4 2y 2 .
(a) Find and classify all xed points of the system
f
x
f
y =
y
x =
then sketch the phase portrait for this system. (Hin
MAT 431 - WINTER 2013
Homework 4
Ryan Szypowski
Due March 7, 2013
1. Prove that the system
x = x y x(x2 + y 2 )
y = x + y y(x2 + y 2 )
has a closed orbit for any > 0 using the Poincar-Bendixson theorem. It may be
e
easiest to convert this problem to polar
MAT 431 - WINTER 2012
Homework 1 Solutions
Ryan Szypowski
Due January 24, 2013
1. For each of the following, nd and classify all xed points. Use both linear stability
analysis (in the cases where it works), as well as a graphical approach.
(a) x = (x + 1)
MAT 431 - WINTER 2012
Homework 2
Ryan Szypowski
Due Feb 5, 2013
1. For each of the following linear systems, classify the single xed point at the origin.
Then, sketch a few solution curves in the phase plane. Be sure to include the invariant
stable and un
MAT 431 - WINTER 2013
Homework 2 Solution
Ryan Szypowski
Due Feb 5, 2013
1. For each of the following linear systems, classify the single xed point at the origin.
Then, sketch a few solution curves in the phase plane. Be sure to include the invariant
stab
MAT 431 - WINTER 2012
Homework 3
Ryan Szypowski
Due Feb 26, 2013
1. Let f : R2 R be dened by
f (x, y) = x4 2x2 + y 4 2y 2 .
(a) Find and classify all xed points of the system
f
x
f
y =
y
x =
then sketch the phase portrait for this system. (Hint: We are
MAT 431 - WINTER 2012
Homework 1
Ryan Szypowski
Due January 24, 2013
1. For each of the following, nd and classify all xed points. Use both linear stability
analysis (in the cases where it works), as well as a graphical approach.
(a) x = (x + 1)x(x 1)
(b)