**INTRODUCTION TO**
**NONLINEAR DYNAMICS AND CHAOS**

Physics
D60-0 |
Spring quarter 1999 |

TTh 2:30-4:00 Tech M120 |
Predrag
Cvitanovic' |

Course description

Project descriptions

Related courses, software

**TEXT:
**
Steven H. Strogatz:
*Nonlinear
Dynamics and Chaos* (Addison-Wesley 1994). All chapter and exercise
numbers refer to this book, unless stated otherwise.

Some material and problem sets from
*Classical and Quantum Chaos*
webbook, available on
www.nbi.dk/ChaosBook/

For people following the course, check the e-mail list. Please subscribe to the course e-mail distribution - send e-mail with text (and no header):

To: info@complex.nbi.dk

Subject: <EMPTY>

subscribe chaos_course

Please deliver solutions to problem sets by Tuesday, at the lecture, or place them in Yueheng's mailbox, Physics and Astronomy Department mailroom Tech F223.

**Lecture 1 ***2:30
- 4:00 tuesday, March 30, 1999 in Tech M120*
**Preview;
Basic properties of differential equations**

A brief historical introduction to the subject.
`Reading: chapter 1;
chapter 2, sections 2.1-2.3`
`Optional reading:
Historical appendix
of the webbook might be amusing reading.`
`Problems:
2.1.1, 2.1.2, 2.1.3,
2.2.7, 2.2.11
`

**Lecture 2 ***2:30
- 4:00 thursday, April 1, 1999 in Tech M120*
**One-dimensional dynamics;
Bifurcations**
`Reading: chapter 2, chapter 3, sections 3.0-3.3 `
`Optional reading:
Sect 4.1
of the webbook,
for a more abstract definition of dynamical systems.
`
`Problems: 2.4.7, 2.6.1, 2.7.3, 2.8.6
`

**Lecture 3 ***2:30
- 4:00 tuesday, April 6, 1999 in Tech M120*
**Bifurcations**
`Reading: all of chapter 3`
`Problems: 3.1.2, 3.1.3, 3.2.2, 3.4.4, 3.5.4, 3.6.5
`

**Lecture 4 ***2:30
- 4:00 thursday, April 8, 1999 in Tech M120*
**Bifurcations**
`Reading: chapter 3`

**Lecture 5 ***2:30
- 4:00 tuesday, April 13, 1999 in Tech M120*
**Flows on the circle**
`Reading: chapter 4, sections 4.0-4.4`
`Problems: 4.1.2, 4.3.1, 4.3.2 `

**Lecture 6 ***2:30
- 4:00 thursday, April 15, 1999 in Tech M120*
**Two-dimensional systems**
`Reading: Chapter 5`
`Problems:
5.1.1, 5.1.9, 5.2.1, 5.2.6, 5.2.13
`

**Lecture 7 ***2:30
- 4:00 tuesday, April 20, 1999 in Tech M120*
**Conservative systems and index theory**
`Reading: chapter 6`
`Problems: 6.1.2, 6.1.10, 6.3.2, 6.3.10, 6.5.8-10, 6.5.19
`

**Lecture 8 ***2:30
- 4:00 thursday, April 22, 1999 in Tech M120*
**Conservative systems and index theory**
`Reading: chapter 7`
`Problems:
7.1.9, 7.2.9`

**Lecture 9 ***2:30
- 4:00 tuesday, April 27, 1999 in Tech M120*
**Poincaré-Bendixson theorem**
`Reading: chapter 7`
`Problems:
7.3.1`

**Lecture 10 ***2:30
- 4:00 thursday, April 29, 1999 in Tech M120*
**Non-linear oscillators and averaging**
`Reading: chapter 7`
`Problems:
7.5.3, 7.6.13`

**Lecture 11 ***2:30
- 4:00 tuesday, May 4, 1999 in Tech M120*
**Bifurcations in two-dimensional systems**
`Reading: chapter 8`
`Problems: 8.1.4, 8.1.11, 8.2.1, 8.2.5`

**Lecture 12 ***2:30
- 4:00 thursday, May 6, 1999 in Tech M120*
**Bifurcations in two-dimensional systems**
`Reading: chapter 8`
`Optional reading (not required in the course):
the
damped driven pendulum, Tomas Bohr's notes.
`
`Problems: 8.3.1, 8.7.3`

**Lecture 13 ***2:30
- 4:00 tuesday, May 11, 1999 in Tech M120*
**Lorenz equations**
`Reading: chapter 9, sections 9.0, 9.2`
`Why
Dame Mary L. Cartwright discovered chaos in 1940's.
`
`Problems: 9.2.2, 9.2.4, 9.2.6, 9.4.2
(J.M.: 9.1.4, 9.2.2)
`

**Lecture 14 ***2:30
- 4:00 thursday, May 13, 1999 in Tech M120*
**Lorenz system, one-dimensional maps**
**Reading**: chapter 9, sections 9.3-9.5;
chapter 10, sections 10.0-10.4**Optional reading**:
Sect 4.2
of the webbook,
stability/Jacobinas of flows.
**Problems**:
9.2.4, 9.3.9, 9.4.2,
10.1.9
**Projects**: the current term projects are listed on
*www.nbi.dk/ChaosBook/projects/*;
please get in touch with me soon if you have a project proposal.
**Final exam**: current plan is to give you a small project-length
problem Friday morning June 4. The problem would involve analysis of a
dynamical system (fixed points, stability, bifurcations) and numerical
experimentation with integration of a set of differential equations
describing the system. Due date would be 2:30 PM
June 8, 1999, my office.

**Lecture 15 ***2:30
- 4:00 tuesday, May 18, 1999 in Tech M120*
**One-dimensional maps**
`Reading: chapter 10, sections 10.5-10.7`
`Problems:
10.1.12, 10.1.13, 10.2.6, 10.4.5
`

**Lecture 16 ***2:30
- 4:00 thursday, May 20, 1999 in Tech M120*
**Renormalization theory; fractals**
`Reading: chapter 10, sections 10.6-10.7; chapter 11`
`Problems:
10.7.4, 11.3.3, 11.4.6
`

**Lecture 17 ***2:30
- 4:00 tuesday, May 25, 1999 in Tech M120*
**Strange attractors**
`Reading: chapter 12`

**Lecture 18 ***2:30
- 4:00 thursday, May 27, 1999 in Tech M120*
**Chaotic averaging: escape rates**

`Reading period starts`

**Lecture 19 ***2:30
- 4:00 tuesday, June 1, 1999 in Tech M120*
**Chaotic averaging: escape rates?**
`Problems: we concentrate on the term paper projects`

**Lecture 20 (the last of the course)
**

A moment of truth; exhaustion has sat in, so we cancelled the lecture.

The exam involves analysis of a dynamical system (fixed points, stability, bifurcations) and numerical experimentation with integration of a set of differential equations describing the system.

If you are working on your own project, but have discovered that it is too ambitious for this course (it is enough to work through somebody's good published paper, one does not need to produce a piece of frontline research), and you are not satisfied with your progress, this is the last day to give up on the project, and take the exam instead.

Alernative:

**
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