Introduction to
Nonlinear Dynamics and Chaos

Instructor

Roman Grigoriev
Office: Howey W304
Phone: (404) 385-1130
E-mail:
Office hours: Tuesday 2-3pm

Tuesdays and Thursdays, 9:35-10:55am
Room S204, Howey Physics Building

The material covered includes differential equations, their stability and bifurcations, iterated maps, deterministic chaos, fractals, and strange attractors with applications to physical, chemical, and biological systems.

Steven Strogatz: Nonlinear Dynamics and Chaos (Perseus Books, 1998). All chapter and exercise numbers refer to this book, unless stated otherwise.
Other books you might find useful:
• Maps:
  • H. Schuster, Deterministic Chaos, VCH, Weinheim.
  • K. Aligood, T. Sauer, J. Yorke, Chaos: an Introduction to Dynamical Systems, Springer-Verlag, New York.
• Hamiltonian Systems:
  • A. Lichtenberg, M. Liebermann, Regular and Stochastic Motion, Springer-Verlag, New York.
• Perturbation Theory:
  • C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scienists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, New York.

Homework assignments will be posted on the web every Thursday and will be due next Thursday in class. You can discuss problems with each other, but the solutions have to be executed and submitted individually. Some assignments will include problems (in blue font), which are mandatory for graduates and optional (for extra credit) for undergraduates. In general you are expected to comply with the academic honor code. Grades will based on the results of the homework assignments (50%), midterm (20%), and final exam (30%).

January 6
1. Introduction
Reading: Chapter 1, lecture notes

January 8
2. Flows on the line
Reading: Chapter 2, lecture notes
Fun stuff: snowflakes and synchronization
Problem set #1: 2.1.5, 2.2.10, 2.3.3, 2.4.2, 2.4.9 (solutions)

January 13
3. Lyapunov function
Reading: Chapter 2, lecture notes

January 15
4. Numerical solution of nonlinear ODEs
Reading: Chapter 2, lecture notes, Sections 16.0-16.3 of Numerical Recipies by Press et al.
Problem set #2: 2.5.3, 2.7.6, 2.8.3 (solutions)
Note: There is a typo in 2.8.3 part (c): you are to plot ln(E) vs. ln(Delta t). Explain how the slope is related to the order of the method.

January 20
5. Bifurcations in one-dimensional systems
Reading: Chapter 3, lecture notes

January 22
6. Bifurcations in one-dimensional systems (continued)
Reading: Chapter 3, lecture notes
Problem set 3: 3.1.3, 3.2.6, 3.3.1, 3.4.8, 3.5.7 (solutions)

January 27
7. Imperfect bifurcations
Reading: Chapter 3, lecture notes

January 29
8. Flows on a circle
Reading: Chapter 4, lecture notes
Problem set 4: 3.6.2, 3.7.6, 4.1.8, 4.3.7, 4.6.3 (solutions)

February 3
9. Two-dimensional systems
Reading: Chapter 5, lecture notes

February 5
10. Phase plane analysis
Reading: Chapter 6, lecture notes
Problem set 5: 5.1.10, 5.2.13, 6.1.3, 6.2.1, 6.3.1, 6.4.7 (solutions)

February 10
11. Conservative systems
Reading: Chapter 6, lecture notes

February 12
12. Index theory
Reading: Chapter 6, lecture notes
Problem set 6: 6.5.10, 6.6.3, 6.7.2, 6.8.5, 6.8.7 (solutions)

February 17
13. Limit Cycles
Reading: Chapter 7, lecture notes

February 19
14. Perturbation theory
Reading: Chapter 7, lecture notes
Problem set 7: 7.1.6, 7.2.9, 7.3.9, 7.4.1 (solutions)

February 24
15. Perturbation theory
Reading: Chapter 7, lecture notes, Bender and Orszag

February 26
16. Nonlinear oscillators and averaging
Reading: Chapter 7, lecture notes, Bender and Orszag

March 3
Mid-term exam

March 5
17. Bifurcations in two dimensions
Reading: Chapter 8, lecture notes
Problem set 8: 7.5.4, 7.6.3, 7.6.6, 7.6.17 (solutions)

March 10
18. Hopf bifurcation
Reading: Chapter 8, lecture notes

March 12
19. Global bifurcations of cycles
Reading: Chapter 8, lecture notes
Problem set 9: 8.1.8, 8.1.11, 8.2.1, 8.2.9, 8.3.1 (solutions)

March 24
20. Quasiperiodicity and Poincare maps
Reading: Chapter 8, lecture notes

March 26
21. Floquet Theory
Reading: Chapter 8, lecture notes
Problem set 10: 8.4.2, 8.4.12, 8.5.2, 8.6.1, 8.7.5 (solutions)

March 31
22. Lorenz equations
Reading: Chapter 8, lecture notes
Numerical exploration of different dynamical regimes (Maple)

April 2
23. Lorenz equations
Reading: Chapter 9, lecture notes
Problem set 11: assignment (solutions)

April 7
24. One-dimensional maps
Reading: Chapter 10, lecture notes

April 9
25. Universality
Reading: Chapter 10, lecture notes
Matlab simulations of the Rossler system: reduction to 2D and 1D maps and stretching of phase space volumes
Problem set 12: 10.1.11, 10.2.6 (see comments for 10.2.3), 10.3.11, 10.4.1

April 14
26. Statistical characterization of chaotic motion
Reading: Chapter 10, lecture notes, Schuster

April 16
27. Fractals
Reading: Chapter 11, lecture notes, Schuster
Fractals in nature, biology, and mathematics.
Problem set 13: assignment (solutions)

April 21
28. Fractals
Reading: Chapter 11, lecture notes, Schuster

April 23
29. Strange Attractors
Reading: Chapter 12, lecture notes, Schuster

April 29 (8-10:50am)
Final exam

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