ES_APPM 322 Homework 4 Solutions
1. Classification of Fixed Points
Identify the type of fixed point and sketch the phase portrait for linear system x = Lx for the 3 cases
1 2
0
2
1
2
L1 =
L2 =
L3 =
1
2
1
5
1
3
Tr
Det
Tr2
4Det
L1
L2
L3
3
5
2
4
2
7
1
17
8
S

322 Applied Dynamical Systems
Syllabus
Class Information
Lectures are TuTh 3:30-5 in M416 (applied math conference room)
Professor: Hermann Riecke, Office Tech M458, h-riecke@northwestern.edu
TA: Cangjie Xu, Office Tech M463, cangjiexu2013@u.northweste

Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 4
Thursday, April 21, 2016
due Thursday, April 28, 2016
1. Classification of Fixed Points
Identify the type of fixed point and sketch the phase portrait for the linear system
x = Lx
for

Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 1
Thursday, March 31, 2016
due Thursday, April 7, 2016
The numbered problems are taken from Nonlinear Dynamics and Chaos by S.H. Strogatz
(2nd edition).
1. Fixed Points and Stability: 2

Applied Dynamical Systems
Midterm, May 6, 2014
Name:
Spring 2014
Riecke
Score
Instructions:
1. Print your name on this page in the space provided.
2. If you need extra space, use the back of a page
and make a note that you have continued elsewhere.
3. Thi

Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 5
Friday, April 29, 2016
due Tuesday, May 10, 2016
1. Poincare-Bendixson Theorem
Consider the dynamical system
x = y + x3 x x2 + y 2
2
y = x + y 3 y x2 + y
,
2 2
(1)
.
(2)
Perform a lin

Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 2
Thursday, April 7, 2016
due Thursday, April 14, 2016
1. Vector Fields and Bifurcations
For each of the following dynamical systems sketch the bifurcation diagram of the
fixed points a

Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 3
Thursday, April 14, 2016
due Thursday, April 21, 2016
1. Scaling at SNIC
Consider the following dynamical system on a circle
d
= r + sin2n
dt
with n integer. For which value of r doe

Some additional plots for Problem 3, illustrating the dynamics:
Problem 2
(a)
The plots of V and w are for initial conditions at (0, 0).
Different initial conditions converge to the invariant circle.
(b)
Each row shows the resulting phase portrait for a d

Midterm, May 6, 2014, Section: Name: Page 3
Problem 1 (20 points). Consider the one-dimensional dynamical system
with f (at, a) a smooth function in both arguments. In addition, ﬁrm #0) = 0 and
6 f My 7E O for a: = $0 and ,u 2 #0. Give a condition on f(

ES_APPl\/l 322 Homework 1 Solutions
1. Book Exercise 2.2.3 Analyze x = x e x3 graphically.
Solution Fixed points are x’“ = —1,0. 1. Since f(x) > D on interval (—00,—1) U (0.1) and f(x) <
0 on interval (—1,0) U (1,00). fixed points X* = :l:1 are stable a