MATH332: Nonlinear Dynamics and Chaos (2008 D2)

Lectures and Lecturers

Continuous DynamicsMaps and Bifurcations
Wednesday 12:00—14:00, C5C 240
(E7B 209 Math/Phys Computing Lab on 13 August)		 Friday 12:00—14:00, C5A 313
(sometimes in E7B 209 Math/Phys Computing Lab)
Prof. Paul Smith
Office: E7A 402
Ph: 9850 8944
email: pdsmith@maths.mq.edu.au 		Dr. Ross Moore
Office: E7A 419
Ph: 9850 8955
email: ross@maths.mq.edu.au
There will be no tutorials.
Some lectures, in particular on Fridays, may be held in the Math/Phys Computing Laboratory: E7B 209.

Additions &/or corrections will be made to this webpage throughout the semester.

Please contact us if you have a query or a correction.

Review material for the "Maps & Bifurcations" half-unit:

Bifurcations of Interval Maps
Stable and Unstable Manifolds for ODEs
Poincaré Maps
Lyapunov Exponents

Study Guide

 Download the study guide: (pdf) or (doc)
Past exams: 2006, 2007.

Assignments
There will be eight (8) assignments, four for each half-unit.

Question sheets Due date Instructions
Assignment 1 Tuesday 19 August
(Week 3) 		complete all exercises in the 3 notebooks:
Function Iteration 1 , 2 , 3 using Mathematica
Assignment 2 Friday 12 September
(Week 6) 		Solutions
Assignment 3 Tuesday 16 September
(Week 7) 		Use Mathematica (or other software tools) to obtain an estimate for the Feigenbaum number, to as many decimal digits of accuracy as you are able. Determine also the value of λ at the limit of period-doubling for the logistic map, as accurately as you can.   Provide sufficient evidence to justify your results.
(Here is the PDF that I used in the lecture, and a Mathematica notebook and package that allows you to plot Bifurcation diagrams.)
You may also wish to research other methods of calculating the Feigenbaum number.
Assignment 4 Tuesday 7 October
(Week 8) 		Solutions, Graphs
Assignment 5 Tuesday 14 October
(Week 9) 		This is quite long, but none of the exercises requires any sophisticated mathematics beyond 1st-year level.
As an optional extra, use Mathematica to illustrate your answers; in particular, reproduce the graphics shown on the question sheet.
Solution Notebook
Assignment 6 Monday 3 November
(Week 12)		 Solutions
Assignment 7 Tuesday 11 November
(Week 12)		 Work by yourself through the ideas in this assignment sheet, before these are discussed in lectures.
Solution Notebook
Assignment 8 Tuesday 11 November
(Week 13)		 Solutions
(Pick up marked papers from Maths Dept. office: E7A-418.)

Notes, Supplementary Reading and Links for the "Maps and Bifurcations" half-unit

Week 1

Here are some references that were used in the 2006 offering of this unit. They make for interesting background reading. Many of the ideas they contain will be discussed in greater detail throughout this unit.

F. Diacu, The Solution of the n-body Problem, The Mathematical Intelligencer, 1996, 18, pp. 66—70. (pdf)

P.E. Rapp, T.I. Schmah, A.I. Mees, Models of knowing and the investigation of dynamical systems, 1999, Physica 132D, pp. 133—149. (pdf)

Wang Sang Koon, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross, The Genesis Trajectory and Heteroclinic Connections, (AAS 99-451), Astrodynamics 1999, AAS Vol. 103, Part III, 2327—2343. (pdf)

John Baez on: The 3-body problem, libration points (a.k.a. Lagrange points) and the interplanetary superhighway (html)

Takashi Kanamaru's double pendulum simulation (html with Java applet)

I. Stewart, Does God Play Dice? The New Mathematics of Chaos, 1989, Penguin.

S. Elaydi, Discrete Chaos, 2000.

R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., 1989, Addison-Wesley.

powered by Mathematica Mathematica Demonstrations: (download the free Mathematica Player software)
Dynamic Systems Theory, Chaos Theory, Differential Equations, Physics.

Java applets that can be run online from the CACTUS Group, Simulation lab page:
Logistic Map and Bifurcation Diagram

Week 2

Some simple notebooks for getting used to using Mathematica and which explore Recurrences and Function Iteration.

Week 3 (2006)

Partial notes, as a Mathematica notebook, requiring a package GraphicalAnalysis.m .

Week 4 (?)

A good article which explains many of the features visible in the bifurcation diagram of the Logistic map:
Richard D. Neidinger, R. John Annen; The Road to Chaos is Filled with Polynomial Curves, The American Mathematical Monthly, Vol. 103, No. 8. (Oct., 1996), pp.640–653. (pdf)

Week 6 (?)

Read the classic article: T.-Y. Li, J. A. Yorke, Period three implies chaos, 1975, The American Mathematical Monthly, Vol. 82, No. 10. (Dec.), pp. 985–992. (pdf)

Week 9

The Mathematica notebook used in the Friday lecture of Week 9.

Week 11

Here is the Mathematica Notebook on the Hénon map, that was used in the lecture.

Week 12

Here is the Mathematica Notebook on toral maps, that was used in the lecture.