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Introduction to
Nonlinear Dynamics and Chaos
svn: $Author: domenico $  $Date: 20060504 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 2.12.3
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 onedimensional systems

Reading: Chapter 3
 January 19
4. Bifurcations in onedimensional 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. Twodimensional 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. Twodimensional 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 6.76.8
 February 23
 administered by R. Grigoriev
14. Midterm exam

(solution, part 1 by Domenico Lippolis)
 February 28
15. Limit cycles 
Reading: Chapter 7, sections 7.07.3
 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 2d 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: extrapoints 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: extrapoints 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 2024
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: extrapoints problem for everybody, catchup 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: extrapoints 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: extrapoints problems for everybody]
(solutions by Rytis Paskauskas)
 April 18
27. Onedimensional 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: extrapoints 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)
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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