[ next lecture | course info | description | FAQs | ChaosBook | projects | webbook help | Cvitanović schedule ]

Introduction to
Nonlinear Dynamics and Chaos

PHYS 4267/6268  -  Spring 2006
www.cns.gatech.edu/~predrag/phys4267
TTh 9:35-10:55 Howey S204

svn: $Author: domenico $ - $Date: 2006-05-04 15:21:55 -0400 (Thu, 04 May 2006) $
-

Course Schedule
January 10
1. A brief history of motion in time
Reading: Chapter 1; Chapter 2, sections 2.1-2.3
Optional reading: ChaosBook.org Brief history of chaos might amuse you.
January 12
2. Vector fields and flows
Reading: Chapter 2
Problem set 1: 2.1.1, 2.1.2, 2.1.3, 2.2.7, 2.2.11, 2.4.7, 2.6.1 (solutions)
January 16
Institute holiday - MLK Day
January 17
3. Bifurcations in one-dimensional systems
Reading: Chapter 3
January 19
4. Bifurcations in one-dimensional systems
Reading: Chapter 3
Problem set 2: 2.7.3, 2.8.6, 3.1.2, 3.2.2, 3.4.4, 3.5.4 (solutions)
January 24
5. Bifurcations in the presence of symmetry
Reading: Chapter 3
January 26
6. Imperfect bifurcations
Reading: Chapter 3
Problem set 3: 3.4.11, 3.5.8, 3.7.5, 3.4.12 (solutions by Daniel Borrero)
January 31
7. Flows on the circle
Reading: Chapter 4
February 2
8. Two-dimensional systems
Reading: Chapter 5, sections 5.1, 5.2
Problem set 4: 4.1.5, 4.3.1, 4.4.1, 5.1.1, 5.1.4, 4.3.2, 4.5.3 (solutions by Chris Malec)
February 7
9. Two-dimensional systems
Reading: Chapter 5, section 5.3
February 9
10. Phase plane analysis
Reading: Chapter 6, sections 6.1, 6.2, 6.3
Problem set 5: 5.1.9, 5.2.2, 6.1.2, 6.3.10, 6.1.14 (solutions by Daniel Borrero)
February 14
11. Phase plane analysis
Reading: Chapter 6, sections 6.4, 6.5, 6.6
February 16
12. Conservative Systems
Reading: Chapter 6, section 6.5
Problem set 6: 6.3.12, 6.5.8, 6.5.9, 6.5.10, 6.5.12, 6.5.19 (solutions by Chris Malec, phase portrait from problem 6.5.12, problem 6.5.10 )
February 21 - lecture by R. Grigoriev
13. Pendulum, index theory
Reading: Chapter 6, sections 6.7-6.8
February 23 - administered by R. Grigoriev
14. Midterm exam
(solution, part 1 by Domenico Lippolis)
February 28
15. Limit cycles
Reading: Chapter 7, sections 7.0-7.3
March 2
16. Relaxation oscillators
Reading: Chapter 7, section 7.5
Optional reading: ChaosBook.org chapter Get straight, section 7.3 illustrates simplification of a mechanical dynamical system by linear scalings and nonlinear time and space reparametrization.
Problem set 7: 6.7.4, 6.8.12, 7.1.6, 7.2.6, 7.5.3; the 2-d system part 2 of the midterm exam (solutions by Danny Caballero, problem 6.8.12.c )
March 7
17. Nonlinear oscillators and averaging
Reading: Section 7.6
March 9
18. Nonlinear oscillators and averaging
Reading: Section 7.6
Problem set 8: 7.5.1, 7.6.12, 7.6.14, 7.5.7, 7.6.2, 7.6.25 [pink: extra-points problem for everybody] (solutions by Danny Caballero)
March 14 - lecture by R. Grigoriev
19. Bifurcations in two dimensions
Reading: Chapter 8, sections 8.1, 8.2
March 16 - lecture by R. Grigoriev
20. Hopf bifurcation
Reading: Chapter 8, sections 8.2, 8.3, 8.4
Problem set 9: 8.1.4, 8.2.3, 8.4.3, 8.1.11, 8.2.9, 8.3.1 [pink: extra-points problem for everybody] (solutions , problem 8.4.3 , Problem 8.2.3: we can see from Matthew Massengill's plots (mu less than 0 , mu=0 , mu>0 ) that, no matter what happens, there is never a stable limit cycle enclosing the region of interest, therefore the Hopf bifurcation cannot be either sub- or supercritical, it has to be degenerate.)
March 20-24
Midterm recess
March 28
21. Josephson junction/driven pendulum problem
Reading: Chapter 8, section 8.5
March 30
22. Quasiperiodicity and Poincare maps
Reading: Chapter 8, sections 8.6, 8.7
Optional reading (not required in the course): the damped driven pendulum, Tomas Bohr's notes.
Problem set 10: 8.5.1, 8.6.2, 8.7.1,
8.7.A [not Strogatz]: Consider the system
dx/dt = a, dy/dt = b,
where both x and y are defined mod 1.
a) Define a Poincare' section and compute the corresponding Poincare' map.
b) Using the map, determine the type of trajectories for different values of a and b.
8.6.4, 8.7.2, 8.7.6, 8.7.7, 8.7.8, [pink: extra-points problem for everybody, catch-up opportunity] (solutions by Adam Perkins and TA's, problem 8.7.A)
April 4
23. Chaos rules
Reading: material covered in the class is not in Strogatz
Optional reading (not required in the course): How Dame Mary L. Cartwright discovered chaos in 1940's.

April 6
24. Lorenz chaotic attractor
Reading: Chapter 9, section 9.2
Problem set 11: 9.2.1, 9.2.3, 9.2.4 [pink: extra-points problem for everybody] (solutions by Adam Perkins)
April 11
25. Chaos
Reading: Chapter 9
April 13
26. Chaos
Reading: Chapter 9.3, 9.4, 9.5
material covered in the class, but not in Strogatz: today's lecture, related ChaosBook pages,
Play: run R. Grigoriev's matlab simulations of the Rossler system: reduction to 2D and 1D maps and stretching of phase space volumes.
Problem set 12: from ChaosBook - 2.8, 3.1, 4.3; from Strogatz (you can modify Grigoriev matlab codes, or write your own) - 9.3.9, 9.3.10 [pink: extra-points problems for everybody]
(solutions by Rytis Paskauskas)
April 18
27. One-dimensional maps
Reading: Chapter 10.1, 10.2
Optional reading (not required in the course): Universality in chaos (or, Feigenbaum for cyclists), Zakopane School of Theoretical Physics lectures, by P. Cvitanović, Acta Phys. Polonica A65, 203 (1984). These lectures are an introduction to the reprint selection Universality in Chaos (Adam Hilger, Bristol, 1989), with highly readable papers by E.N. Lorenz, M. Henon, R. May, M.J. Feigenbaum, and others.
April 20
28. Universality
Reading: Chapter 10.3, 10.4, 10.6, 10.7
Problem set 13: 10.1.10, 10.1.12, 10.3.5, 10.6.1, 10.6.6, 10.7.4 [pink: extra-points problems for everybody]
(solutions by Daniel Borrero)
April 25
29. Fractals
Reading: Chapter 11.1, 11.2, 11.3, 11.4
Problem set 13 due in class
April 27
30. Strange attractors
Reading: Chapter 12.1, 12.2, 12.3
Friday Apr 28
classes end
Wed, May 3
8:00 - 10:50 in Howey S204: final exam (solutions by Predrag Cvitanović and Domenico Lippolis)
(Closed book, on material covered in lectures 15 to 30)

Course Instructor Opinion Survey
Please fill out the online Course Survey. This is an opportunity to provide feedback regarding the contents of the course, the style and quality of the presentation, or any other aspect of the course. Tell us what you liked and what you did not like. Your input is very valuable and will benefit students taking this course in subsequent years.
The final song: R.E.M.
It's The End Of The World As We Know It (mp3 - courtesy of Stephen Hsu)

-

grades deadline Mon, May 8

-
Predrag Cvitanović