CDS 140b, Spring 2012

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This will be the homepage for CDS 140b for Winter 2012.

Introduction to Dynamics


Teaching Assistant

  • Katie Broersma
  • Office hours: Monday 1-2pm, Steele House Conference Room / Steele House 101

Course Description

CDS 140b is a continuation of CDS 140a. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be five homeworks throughout the semester but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.


  • 22 May 2012: Thursday lecture in 202 Guggenheim at 1pm
  • 30 Apr 2012: Next two weeks on nonlinear control; homework 3 will be posted no later than Thursday
  • 20 Apr 2012: Homework 2 is posted.
  • 4 Apr 2012: Starting 5 Apr 2012, New Class Timing T, Th 9 AM - 10:25 AM, 314 ANB
  • 17 Mar 2012: At least one student has a conflict with the class time; we will see if we can find a time compatible with everyone during the first week of the quarter.
  • 17 Mar 2012: Doug MacMynowski is traveling the first few weeks of the quarter; direct all questions to Shaunak...
  • 17 Mar 2012: web page creation: the list of topics below has not yet been edited from last year; please ignore!

Lecture Schedule

Week Date Topic Suggested Reading/Lecture Notes Homework
1 3 Apr
5 Apr
Perturbation Theory
  • Regular Perturbation
  • Poincare-Lindstedt Method
  • Periodically Perturbed Systems
  • Khalil, 10.1-10.3
  • Strogatz, 7.6
  • Verhulst, 9.1, 10.1
  • Chapters 1 & 6 in Dynamical Systems, the Three-Body Problem, and Space Mission Design, Wang Sang Koon, Martin W. Lo, Jerrold E. Marsden, Shane D. Ross, ISBN 978-0-615-24095-4.

Homework 1 Solution 1

2 10 Apr
12 Apr
Averaging Method
  • Periodic Case
  • Periodic Solutions
  • General Case
  • Khalil, 10.4-10.6
  • Strogatz, 7.6
  • Verhulst, 11.1-11.3, 11.8
  • Coupled oscillator models with no scale separation, Philip Du Toit, Igor Mezić, Jerrold Marsden, Physica D 238 (2009) 490–501.
3 17 Apr
19 Apr
Singular Perturbations
  • Finite Interval
  • Infinite Interval
  • Stability Analysis
  • Khalil, 11.1-11.3, 11.5
  • Chapter 9, Mathematical Biology: I. An Introduction, J. D. Murray, Springer; 3rd edition.

Homework 2 Solution 2

4 24 Apr
26 Apr
No class this week
5 1 May
3 May (Mid-term period)
Nonlinear control I
  • Overview of techniques
  • Controllability and Lie brackets
  • Gain scheduling
  • Feedback linearization
  • Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
  • Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
  • Lecture notes

Homework 3 Solution 3

6 8 May (Mid-term period)
10 May
Nonlinear control II
  • Backstepping
  • Sliding mode control

Homework 4 Solution 4

7 15 May
17 May

Lagrangian Coherent Structures

LCS lecture notes
8 22 May
24 May
Turbulence and non-normal growth
9 29 May
31 May
  • Tuesday: Chaos
  • Thursday: Final Project Presentations
  • "Perturbation and stability analyses of bubble dynamics in a linear viscoelastic (tissue-like) medium", C. Hua
  • Overview of "Multiple Lyapunov Functions and other analysis tools for switched and hybrid systems", S. Livingston
  • "Bifurcations in two layer model of ice albedo feedback", R. Wills

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Course Textbooks

  • H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
  • S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
  • F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

Additional Sources:

  • L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777


Lecture notes:

A brief description of the mathematical concepts presented during the lectures may be posted as lecture notes. These should assist students with the mathematical concepts presented during the lecture. Complete lecture notes will not be posted.

Collaboration Policy

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reflect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

Grading Policy

The final grades will be evaluated based on homework assignments (5*15%=75%) and final projects (25%).

Late Homework

Each student is allowed one late day which means only one homework assignment may be handed in up to one day late. Other than this day, late homework will not be accepted. Exceptional circumstances (such as medical situations) with appropriate documentation will be considered by the instructors.


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