Dynamical Systems
and Chaos

110.421, ThF 2-3.15 Bloomberg 168
Dr Mark Haskins
E-mail address: mhaskin@math.jhu.edu
Telephone: 410-516-4047
Department: Mathematics
Office: Krieger 312
Office Hours: TBA
TA: Jason Metcalfe

Homework assignments

HW 0 (review) : due beginning of class 02/01
HW 1: due 02/07 Postscript, and PDF versions available
HW 2: due 02/14 Postscript, and PDF versions available
HW 3: due 02/21 Postscript, and PDF versions available
HW 4: due 02/28 Postscript, and PDF versions available
HW 5: due 03/07 Postscript, and PDF versions available
HW 6: due 03/14 Postscript, and PDF versions available

(Partial) Homework solutions

HW 0
HW 1
HW 2 p1, HW 2 p2, HW 2 p3, HW 2 p4, HW 2 p5, HW 2 p6
HW 3 p1, HW 3 p2, HW 3 p3, HW 3 p4, HW 3 p5, HW 3 p6
HW 4 p1,
HW 5 p1, HW 5 p2
HW 6 p1, HW 6 p2, HW 6 p3, HW 6 p4, HW 6 p5,
HW 7 p1, HW 7 p2
HW 8 p1
HW 9 p1, HW 9 p2, HW 9 p3
HW 10 p1, HW 10 p2, HW 10 p3 HW 10 p4

Matlab Scripts

Here are some Matlabs script you can download:
1. graphical analysis for a user-defined function of one variable.
2. ode plotter for odes of one variable.
3. orbit analysis for orbit analysis of the quadratic maps (for use in Chapter 8).
4. ode plotter for odes of 2 variables.
5. feigenbaum script to find the Feigenbaum numbers (and an auxiliary script needed to run the main script).
6. Filled Julia set plotter..
7. Julia set plotter
8. Coloured Filled Julia set
9. Mandelbrot set (B&W)
10. Mandelbrot set (colour)

About the course

This course is designed to introduce students from a variety of mathematical, scientific and engineering backgrounds to some of the fundamental notions of nonlinear dynamics. We will concentrate on the simplest dynamical systems which can exhibit so-called chaotic behaviour -- the discrete dynamical systems which arise from iterations of real or complex valued functions. The perspective taken in dynamical systems is to attempt to understand the qualitative behaviour of a whole system or classes of systems rather than writing down particular explicit solutions.

The aim is to cover most of Devaney's book and to end the course with a detailed discussion of the well-known Mandelbrot set and to explain what the significance of figures like the one at the top left of this page is.

We will start by introducing basic concepts in dynamical systems in the context of iterations of real valued functions. Even in some of the simplest nonlinear functions like quadratic functions surprisingly complicated phemonena can occur. After a detailed study of iterations of real functions and especially quadratic maps we play the same kind of games with complex valued functions. It is in this context that the Mandelbrot set eventually appears. Fundamental questions about this set remain unanswered. We will try to mention a few of these.

Computers can be an effective tool for "experimentally" discovering properties of dynamical systems, especially discrete ones, and can lead to theoretical discoveries too. The course will include homework that involves computer work. The primary software package we will use is Matlab. A key issue will be to determine when we can rely on the computer results i.e. when does the computer lie?


Confusingly, Robert Devaney has written two different introductory books on chaotic dynamical systems

1. A First Course in Chaotic Dynamical Systems: Theory and Experiment
2. An Introduction to Chaotic Dynamical Systems

The second book is somewhat more advanced than the first. The bookstore has copies of the first title and we shall use this book. The other book is useful if you want to see some slightly more advanced topics on similar material. Other texts you might find helpful for a broader perspective are:

Encounters with Chaos, Denny Gulick
Chaotic Dynamics: an introduction, G.L.Baker and J.P.Gollub
Understanding Nonlinear Dynamics, Daniel Kaplan and Leon Glass
Chaotic & Fractal Dynamics: An Introduction to Applied Scientists & Engineers, F.C. Moon.

You can come and browse any of these books in my office during office hours.


The formal prerequisites for the class are Calculus III and Linear Algebra. Note: a course in differential equations is not a prerequisite.

We will make more extensive use of notions from calculus than from linear algebra. Students who have not taken classes recently that use calculus would benefit by reviewing the basic notions from Calculus I & II.

Grading Policy

Homework, 30%: Assigned weekly and due weekly. Some problems may not be graded. No late homework accepted without a doctor's note. You may consult with classmates but be sure to do most of the work yourself. Assignments will generally be due at the beginning of class on Thursday.

Midterm Exam, 20%: A combination in-class/take-home exam to be given in the week following Spring Break. Further details will be announced.

Project, 20%: An opportunity to study in greater depth material of interest to you and related to the course subject. The project could be theoretical, practical (e.g. building a physical model of something), computer-related (e.g. simulating a system) or some combination of these. This project will be due before the last day of class. The subject of the project will be chosen by you with guidance given by me. Further details will be given later.

Final Exam, 30%: Another combination in-class/take-home exam. Provisionally, in-class component given on final class day, Monday December 11 with take-home part given during preceding weekend.

Computing Information

The Matlab homepage at Mathworks has documentation about all aspects of Matlab. You may find Getting Started helpful if you have never used Matlab before.

Project Information

A number of ideas for projects for the class and project guidelines from the class from Fall 2001.

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Last updated 4 Feb 2002
Mark Haskins