Introduction to
Nonlinear Dynamics and Chaos

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

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)

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