AMS 146 Syllabus - Winter 2007

General Information
Class and Exams Schedule
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General Information

• Textbook: Nonlinear Dynamics and Chaos by Steven H. Strogatz. Publisher: Westview Press.
• Supplemental textbook: A first course in chaotic dynamical systems by Robert L. Delaney. Publisher: Addison Wesley
• Suggested reading (global popular science overview):
  
• The  essence of Chaos by Edward N. Lorenz. Publisher: University of Washington Press
• Chaos: Making a new Science by James Gleick. Publisher: Penguins Book
• The Beauty of Fractals by Heinz-Otto Peitgen and Peter H. Richter. Publisher: Springer-Verlag
  

• Eligibility:
• AMS 27, or MATH 27, or MATH 21 and 24.
  

• Homework: Suggested homework will be given every week and answers will be discussed in Section. You are strongly advised to attempt and complete as much of the homework as possible and go to section to find out the correct answers. Homework is not graded.
  

• Quizzes: Will be held approximately every two weeks and will usually be based on one of the homework problems set during the previous two weeks.
  

• Exams: There will be one mid-term exam and a final.
  

• Grading Policy:
  
• Quizzes: 30 % of total grade.
• Mid-term: 30 % of total grade.
• Final: 40 % of total grade.
  

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Tentative Schedule: (this will be updated as the course progresses)

• Week 0:
  
• Jan 5: General introduction to Dynamical Systems and Chaos

• Week 1:
•  Flows on the line (Chapter 2)
• Bifurcations part 1: Saddle-nodes and transcritical bifurcations. Normal forms. (Chapter 3.1, 3.2)

• Week 2: 
• Bifurcations part 2:  Pitchforks. Example of the bead on a hoop. Imperfect bifurcations. Notion of co-dimension. (Chapter 3.4, 3.6)
• Applications of continuous 1D systems: Population dynamics

• Week 3:
• Introduction to 2D systems. Fixed points and linearization
  
• Solution of linear systems. Classification of fixed points.
  

• Week 4:
• Examples of 2D systems. The nonlinear pendulum, the nonliear damped pendulum and the predator prey model.
  
• Structural stability of fixed points. Example of the predator-prey model. Conservative and reversible systems (Chapter 6.5).

• Week 5:
• Limit cycles (Chapter 7)
• Bifurcations in 2D systems. Example of the solar dynamo

• Week 6: (also based on supplemental textbook)
  
• Midterm
• One-D Maps part 1 (Chapter 10.1,  10.2)
  

• Week 7:
• The Logistic Map in detail;  Universality and Period Doubling route to Chaos. (Chapter 10.4, 10.7)
• Definition of Chaos. Lyapunov exponents; examples. The Tent Map

•  Week 8:
• From dynamical systems to Fractals. Example: the Tent Map. Definition of Fractals.
• 2D maps
  

• Week 9: 
• Fundamental examples of dynamical systems: 1. the restricted  3-body problem
• Fundamental examples of dynamical systems:  2. The Lorenz equations

• Week 10: Further topics if time permits.