CDS 140b, Spring 2011

Introduction to Dynamics
Instructors Nader Motee (CDS) Jun Liu (CDS) Lectures: Tu/Th, 10:30-11:55, 314 ANB Office Hours: By appointment	Teaching Assistant Jerry Cruz Office hours: M 4:30-6:00, 230 ANB

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 the existence of periodic orbits, bifurcation theory, perturbation theory and averaging, advanced stability analysis, 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 CD140b and present a brief review of the paper. The details of the projects will be discussed in the class.

Announcements

• 11 May 2011: Homework #4 posted; due 31 May 2011
• 29 Apr 2011: Homework #3 posted; due 12 May 2011
• 14 Apr 2011: Homework #2 is now posted; due 26 Apr 2011
• 4 Apr 2011: List of Reserves (in SFL) for the course has been created
• 2 Apr 2011: Homework #1 is now posted; due 12 Apr 2011
• 18 Mar 2011: web page creation

Lecture Schedule

Date	Topic	Reading/Lecture Notes	Homework
29 Mar 31 Mar 5 Apr	Limit cycles Recap Ruling out closed orbits Index theory Poincare-Bendixon theorem Lienard systems	Strogatz, 7.0-7.4 Perko, 3.1-3.5, 3.7-3.8, 3.12 Lecture Note 1 Lecture Note 2 Lecture Note 3 Supplementary Lecture Notes	Homework 1 Solutions 1
7 Apr 12 Apr 14 Apr	Stability Theory LaSalle's invariance principle Stability of nonautonomous systems Linearization and converse theorems Boundedness and input-to-state stability Stability of perturbed systems	Khalil, 4.1-4.9, 9.1-9.2 Lecture Note 4 Lecture Note 5 Lecture Note 6 Notes on Stability	Homework 2 Solutions 2
19 Apr 21 Apr 26 Apr	Bifurcation Theory Structural Stability and Peixoto's Theorem Various Types of Bifurcations Higher-Dimensional Bifurcation Unfoldings of Bifurcations	Perko, 4.1, 4.2, 4.3 Lecture Notes 7, 8, and 9	Homework 3 Solutions 3
28 Apr	Guest Lecture	Lecture Slides
3 May 5 May 10 May	Averaging and Perturbation Regular Perturbation Periodically Perturbed Systems Averaging Singular Perturbation Stability Analysis	Khalil, 10.1-10.4, 11.1-10.3, 11.5 Lecture Note 10 Lecture Note 11 Lecture Note 12 Notes on Perturbation	Homework 4
12 May 17 May	Guest Lectures
19 May 24 May	Final Project Presentations

References:

Course Textbooks

• S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
• L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164

Additional Sources:

• H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
• F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346
• S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

Policies:

Lecture notes:

A skeleton file for the lecture notes will be posted the night before each lecture and it will mainly include figures and some text. It is recommended that students come to class with the lecture notes skeleton and use it to fill in the material covered in class. 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*12%=60%), final projects (30%), and participation in class (10%).

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.

Projects:

• Bifurcation Control
• Contraction Analysis (Lingwen)
• Emergent Behavior in Flocks
• Mathematics of Emergence (Hamed)
• Oscillations in I/O Monotone Systems
• Biological Oscillators and Synchronization (Marcella)
• Model Reduction in Biological Systems
• MultiStability and Monotone Systems (Enoch)
• Stability of Switching Systems (Matanya)
• String Stability of Interconnected Systems (Eric)
• Synchronization Using Contraction Theory (Tom)