Dynamical Systems 550.391
Solutions to Homework 3
Joseph Paat
October 4, 2013
1. Follow the instructions in Strogatz, Problem 3.1.1-4 for the system
x=x+
1
r.
x
Solution: Let f (x) = x. In order to nd the critical value, rc , at which the bifurcation occ
Dynamical Systems 550.391
Solutions to Homework 1
Joseph Paat
September 13, 2013
1. Find all of the xed points for the equation x =
f (x) = (1 + sin(2x)/ cos(2x).
1+sin(2x)
cos(2x)
on the real line by nding the zeros of
Solution: Since f is a fraction, it
Dynamical Systems 550.391
Solutions to Homework 2
Joseph Paat
September 20, 2013
1. The general solution of the ODE x = cos(t) is x(t) = sin(t) + C which oscillates on the one
dimensional x-axis. But the text says that one-dimensional systems cannot oscil
Dynamical Systems 550.391
Solutions to Homework 4
Joseph Paat
October 18, 2013
1. Follow the instructions of Strogatz, Problems 4.1.2-7 for the system
= cos() sin(3).
Solution: In order to nd the xed points of the vector eld, we nd where cos() sin(3) = 0
Dynamical Systems 550.391
Solutions to Homework 5
Joseph Paat
October 25, 2013
1. Strogatz 5.1.1
(a) Proof. We are given x = v and v = 2 x. Following the hint in the book, we divide x by v
to get the following separable dierential equation:
dx
x
v
= = 2 .
Dynamical Systems 550.391
Solutions to Homework 6
Joseph Paat
November 8, 2013
1. Follow the instructions of Strogatz, Problem 6.1.1-6 for the system
x = x3 y, y = x + y.
Solution: Solving x3 y = 0 and x + y = 0 for x and y , we nd that the only xed point
Midterm 550.391, Oct. 8, 2013.
Do all of the following three problems. Show all your work. Answers without supporting work may receive no credit.
The exam is open book and open notes. Furthermore, any internet resource may be
used, as long as the URL of t
Midterm 550.391, Oct. 3, 2011.
Do all of the following three problems. Show all your work. Answers without supporting work may receive no credit.
I attest that I have completed this exam without unauthorized assistance from any
person, materials, or devic
Midterm 550.391, Oct. 10, 2012.
Do all of the following three problems. Show all your work. Answers without supporting work may receive no credit.
I attest that I have completed this exam without unauthorized assistance from any
person, materials, or devi