PHYD38
Introduction to Nonlinear Systems and Chaos
A tiny part of the exponential fractal
This page provides current syllabus (below), access to the
lectures notes & recordings, the assignments, and other material.
Notice that Quercus doesnot have all this updated information.
Via Quercus you get announcements & submit homework
(in Quercus/Assignments tab).
This course presents an overview of mathematical analysis of differential
equations ubiquitous in Physical sciences.
Since equations of Physics often contain complicated, nonlinear terms,
you will learn to linearize them around equilibrium solutions.
That will be very useful for quickly establishing the stability (or
instability) of such solutions.
Starting from one-dimensional examples, we proceed to
higher-dimensional cases, where we see interesting new behavior of
dynamical systems emerge, including chaos.
We draw examples of various linear, nonlinear and chaotic
systems from physics and astrophysics, biology and engineering.
The course includes one research project done outside classroom in groups.
Syllabus
TXT file with dates of lectures and exams, topics
Books
Our main textbook is "Nonlinear Dynamics and Chaos, with applications to
Physics, Biology, Chemistry and Engineering" by Steve H. Strogatz (Perseus
Books, 2nd ed., 2015 or 2018). Avoid the 1st edition; there's not much wrong
with it but some topics and exercises are not identical. There is a 3rd edition,
but since I don't use it, you should not use it either - for consistency.
We don't want confusion in page numbers and such. The 3rd edition is not
much improved w.r.t. the 2nd.
You will get advice on how to obtain the textbook in the 1st lecture.
If a chapter from another book will be a required reading, it will be
clearly announced in lectures and mentioned on this page, but it's rare.
Other, not required, books that you may also be interested in are listed at the
end of this page.
Course grading scheme (maximum points)
W = written part, Q = quiz, P = in-class presentation
4 assignments 28% (all) 7% (each)
midterm 18% = 9%(W) + 9%(Q)
final 37% = 19%(W) + 18%(Q)
project 16% = 8%(W) + 8%(P)
activity up to 4%
total not to exceed 100%
Contact and Office Hours
Office hours are right after the lectures (we stay in lecture room if
available or go to my office),
and after tutorials (coffee room on 5th floor
of SW, next to my office 506G).
If you have a question after office hours, send email.
All emails must have a PHYD38 mentioned in the subject & be sent to:
pawel.artymowicz AT utoronto.ca or to pawel AT utsc.utoronto.ca
Lecture recordings
Recordings will be available in Media Gallery of PHYD38 on Quercus or
via this web page (TBD).
Quasiblog/etudes (described below) are also a valuable resource.
□
25 March lecture notes (pdf)
Tutorials
Here is the page
that elaborates on what we did in each tutorial. Useful for study
before exams.
Problem sets - due at 10 am
We will often discuss the solutions of assignments on the due date,
so no late submissions are allowed in this course. Plan ahead -
don't leave the assignment problems until the day before deadline.
Of course there will be occasional emergencies. If you have a medical
or personal emergency, write me an email (and self-declare online in the
former case), and I may be able to either allow a late submission
(if solutions were not presented in the tutorial) or
transfer your points for an assign. set to the final exam.
I suggest that you write the solutions legibly and do the sketches by hand,
then bind snapshots into a pdf file.
Typing your answers would help with the readability but takes more time.
Problems 0 (not graded)
[Since 1st graded assignment will be due at the end of Jan.,
please do the following exercise
but do not submit it - you may volunteer to present your solution at the
tutorial!
We'll discuss the solutions then and try some other numerical methods
on the problem as well.
It's worth doing this exercise now, since a similar one will be part of
the assignment 1 (graded). You'll be able to expand the present solution to
handle other integration methods (and different nonlinear systems). ]
Euler method problem:
Solve dx/dt = 1/x - x starting at t=0 from x(0)=1/2, analytically.
What is the value of x at time t=4?
Then apply the simplest (Euler) numerical method to solve the same problem.
Use any programming language or system you like, but of course it must
be your own program, commented throughout in your own words
(using "AI" for assignments in this course counts as plagiarism).
The accuracy depends on the timestep dt used in calculation.
Study the numerical error, i.e. deviation, called E(n),
of the numerically obtained x(4) from the theoretical value of x(4) for
dt = 10-n,
where n=1,2,3,4,.. (continue until the calculations get unbearably long;
it's up to you to decide what's unbearable, for me it's ~15 min).
Plot or sketch by hand the log-log plot of E(n), that is log|E_n| vs.
log n, and explain the results.
Here is the placeholder for
prob0a-old.py program that you will have create during a tutorial.
Problem set 1,
due on 29.01.24. Cf.
Solutions
Problem set 2,
due on 19.02.24. Published Friday 9 Feb.
See the solutions
page .
Problem set 3,
due on 18.03.24. See the A3 solutions
on this page.
Problem set 4, publ. 20 March.
Due on 4.04.24. See the A4 solutions
on this page.
Note on plagiarism 😞
Plagiarism and cheating do happen, sometimes unwittingly. When I notice
non-sequiturs in your written problem solution, or close similarity of
your solution to fellow student's or any web page, it will be up to you to
convince me that you actually understand what you have written and what I
see is just a coincidence. Looking up solutions on wiki, google, or
using chatGPT and the like plagiarism tools, is increasing.
I'll comment in class how wrong were GPT's answers to our problems 2 & 3
of set A1. (LLMs and their great mimicry of any grammar is more of a
threat to humanities than to hard science educators.)
The university-mandated procedures and/or penalties have to be applied
against plagiarism. Read more in
official docs.
Midterm preparation
Midterm covers chapters 1 to section 7.5 (inclusive) of chapter 7
of Strogatz textbook -- everything that was discussed in Lectures
(cf. recordings in Media Gallery,
Quercus). This means everything up to and including Poincare-Bendixson
theorem, Liapunov function, and Dulac's
criterion (but Dulac for 1 possible question in quiz not for the written
part).
There will be 2 written problems, one short and easy, one a bit longer.
First, for guidance on what kinds of problems may pop up, check out this
file, formatted
like midterm with solutions
As you see, they are like any home assignments or tutorial problems.
Not too complicated.
To prepare well, please choose problems from Strogatz at random
and solve them.
Those with odd numbers are solved in the back of the book, compare your
solution to the solutions provided.
Allowed aids: calculator, handwritten notes
(4 pages on 2 or 4 sheets).
Not allowed: Books, electronic devices, copies/printouts.
Midterm writing started 26.02.2004 at 3:09 and lasted 55 min.
Here is the text of your
midterm with solutions . This will later help you prepare
for the final.
Preparation for the final exam
Date and place in syllabus. 16 April, MW110, 3 hours (9am-12am :-(
Allowed and forbidden aids: same as in midterm.
The only difference: 8 pages of handwritten
notes allowed on 4 or 8 sheets.
Here are very brief comments on our textbook, and other materials
(quasiblog) to help you focus on things that may be seen in the exam.
There will be 4 written problems. None of them will be about specific
topics or particular equations presented in our quasiblog.
About the book. The emphasis in the written part will be on the main
subject areas such as 1-D and 2-D continuous dynamical systems with
all their beautiful behavior and bifurcations (Chapters 3, 5, 6),
discrete systems (iterated maps, chapter 10)
and fractals (chapt. 11). Quiz may be covering main notions from
other chapters as well (those, which we covered in the lectures).
Remember that our textbook has zilllions of problems to practice
your knowledge, answers to odd-numbered problems are in the second part
of the book (2nd ed.).
Please use both even and odd problems in your preparation, though time
is always in short supply, so perhaps a few problems from each chapter
is more realistic than 'zillions'.
Here is an old exam with some quiz questions removed
2014 pdf.
As another training set, here you have exam from 2017 (some Qs removed
from quiz):
2017 pdf.
If you don't see answers to quiz questions in the preparation files, it's on
purpose. Some questions may be repeated in this year's exam.
The Julia set.
2024 exam, solutions to exam problems + quiz questions
Notice that problem 2 had a very (singularly) strong dy/dt next to the vertical axis,
which helps figure out the flow near it. As to the quiz, we didn't really
talk much about Markov chains (no group tackled that!) so I'll ignore 2 quiz
questions while grading (free points!). Here are the JPG's with solutions
prob 1,
prob 2a,
prob 2b,
prob 3,
quiz 1,
quiz 2,
quiz 3.
Preliminary results so far (updated 11 March)
See
results
Quasi-blog
Under this link we discuss:
⋄ Simple numerical schemes
⋄ Theory of flight. Phugoidal oscillations in times of
Zhukovsky, Lanchester, and today
⋄ A dog, a duck and a puzzle of an invisible magnet
⋄ Analytical study of orbit subject to
constant perturbing force (radiation pressure)
⋄ Stability of Lagrange points in R3B
⋄ Simple numerical integrators: importance of
method's order for accuracy
⋄ Numerically solving s'' = s(1-s2) -vq
⋄ Numerical investigation of chaos in Hills equations
⋄ Numerical investigation of chaos in Lorenz dynamical system
⋄ More orbit diagrams. Connection between them and
fractals.
⋄ Research paper on linearized theory of fluids
Student mini-projects
Writeups are to be submitted to the lecturer by email on evening
of 30 March, presentations will take place on 1 April (alternatively,
send the writup on eve.
23 March and present on 25 March, if you prefer).
There will be 4 groups of 4 persons
each.
Everybody gets the same grade, half of the mark is for the quality of
the
writeup, half for presentation. How you divide work on the project between
the members of the group is up to you.
Presenting knowledge of the topic gained from books and
maybe some review or research papers is great, but own calculations
are even more impressive. It's your mini-research after all.
Writeups
PDF (DOC only if you have to) are due 2 days before the meeting, i.e.
on Saturday evening.
Send me the PDF via email. I'll post them on a subpage address
mentioned only in a Quercus announcement, for a few days only.
Everybody should read them before the presentation day and prepare at
least one meaningful question about each project.
Writeups are minimum 7 pages PDF incl. pictures, single spaced i.e. in
printed article style.
The text must have properly cited sources, either books,
articles (review articles are usually best as cited sources for
general readership), or online sources (cite URL).
For example, a text: "... the dynamics described by Newton (1687) ..."
will have a corresponding item in the list of citations at the end:
Newton, I., 1687, Principia Mathematica Philosophie Naturalis, London
Group 1 (Xinfang Zhao, Zac Zhang, Zhen Xu, Abdulrahman D)
smp-Solitons.pdf
Group 2 (Jane Y, Vanessa M, Roger A-T, Pratham P)
smp-Fractals.pdf
Group 3 (Yuang Chen, Byling T, Weishu Sun, Abdilahi Mohamed)
smp-StkMkt-Chaos.pdf
Group 4 (Ivan P, Lineng T, Ethan H, Haoyan Zeng)
smp-Quantum_Chaos.pdf
Presentation
Student miniprojects will be presented in the form of slideshow 25 min long + 5
min discussion on 1 April (or 25 Apr if you prefer - let me know).
Don't forget to have a front page with the title and
names of everybody in the group, at least everybody who wants to get the mark.
We want to hear everybody in the group giving part of the presentation, while
using the same computer with HDMI capability to show us the presentation.
Be aware that some groups have a tendency to go on and on with their
presentation and simply have to be cut off at a certain point.
Practice giving the presentation and time it to avoid such mistakes.
In a real world, it's a useful skill not to exceed allotted time or
other resources.
Topics
1. Lyapunov and his work, a technical presentation on history and applications,
including your own investigation of Lyapunov exponent of some nonlinear systems.
2. Nonlinear filters in digital signal processing
3.
Chaos in economics and the stock market (cf. Puu's book below).
Fair value of options, automatic trading. Room for own analysis of
real stock market data.
4. Markov chains, random walks, applications such as MCMC (Monte Carlo
Markov Chain) and other
5. Power laws & fractals in nature and society.
White, red, gray, brown noise (cf. Schroeder's book below)
6. Nonlinear fluid dynamics, including astrophysical gasdynamics in galaxies
Use
this resource to study the basic physics of nonlinear spiral density waves
(esp. works by Fujimoto and by Roberts).
7. Own study of the exponential
fractal: http://planets.utsc.utoronto.ca/~pawel/iii/fractal.html or other fractals
8. Essential role of nonlinearity in Neural Networks. Machine learning (yes,
the so-called AI). Could be pure literature study. 100% appreciation if
you build a net and teach it some skill, then tell us all about it.)
9. All the interesting things about chaos from Cvitanovic book (below)
10.
Study solitons and their stability using analytical methods. Numerically compute
propagation and interactions of soliton solutions to some equation.
11. Your own idea (pre-approved by professor :-)
12. Quantum chaos, e.g. based on S. Wimberger 'Nonlinear
Dynamics and Quantum Chaos'
Material for some of the last Lectures
Additional topics, beyond those discussed in the Strogatz textbook,
one or more of which may be presented, time allowing.
These and further topic below are all good areas for your own future study.
Topics discussed in lectures will serve as a basis of a couple of quiz questions
in the final exam.
In 2023, I mentioned topics denoted by ☆.
☆ More bifurcation diagrams of discrete mappings
♣ Stability and bifurcations in Engineering
Euler beam buckling as bifurcation
Nonlinear behavior of materials
♣ Nonlinearity, chaos and complexity in Physics and
Astrophysics
☆ The three body and N-body systems
♣ Orbits, Lagrange points, Lyapunov timescales in planetary
and galactic systems
♣ Nonlinear continuum mechanics
Incompressible and compressible fluids
Vortices and turbulence in air and water
☆ Turbulent jets: Chaos out of order and order
in the chaos
♣ Dynamics of galacic and protoplanetary disks
♣ Linear and nonlinear stability and evolution
♣ Nonlinear waves, Fluid resonances, Particle resonances
♣ Nonlinear optics
♣ Quantum chaos
♣ Noise and corruption of signals in physical systems
Noise: white, pink, black, non-power law
Convolution, PSF. Deconvolution.
Wiener & Kalman filters
♣ Chaotic stock market
♣ Solitons: particle-like solitary waves arising in
nonlinear physics
□ Shivamoggi - Nonlinear Dynamics and Chaotic Phenomena,
An Introduction, 2014 (chapter on solitons; among others, connection between
soliton & a homoclinic orbit of a dynamical system)
♣ Nonlinear gas dynamics: examples of astrophysical CFD
(computational gas dynamics)
♣ Speech enhancement. Wiener filters were discussed in SMP.
Yet they are just one of the methods used to denoise the audio.
What are the other methods and how well do they work in practice?
□ Loizou - Speech Enhancement: Theory and Practice, 2013
♣ Researchers continue to study Roessler-like systems with attractors:
□
Ge-DoubleHopfbifurcation-2015.pdf
Double Hopf bifurcation,
as well as multi-dimensional systems with time delays that we haven't studied:
Liu-Six-term_3D_chaotic_system-NonlinSys-2015.pdf.
Notice the Hopf bifurcations, and the literature references to neural networks.
♣ Neural Networks and Computer Intelligence (AI)
□ Rojas - Neural Networks: A Systematic Introduction, 1996
(good exposition w/history)
□ Gershenfeld - The Nature of Mathematical Modeling, 1999
(enormous scope, too brief on NNs)
□ Negnevitsky - Artificial Intelligence. A Guide to Intelligent
Systems, 2004 (v. simple, practical intro)
□ Haykin - Neural Networks and Learning Machines, 2008 (advanced
book)
□ Hinton - Coursera
StG CS course Neural Networks for Machine Learning by Geoffrey Hinton,
UofT & Google, top expert in NNs and AI.
Other recommended books
* Devaney R., "A first course in chaotic dynamical systems (...)" (1992)
2nd/3rd year math course downtown was using it; mathematical but understandable.
* Scheinerman E.R., "Invitation to Dynamical Systems" (PH 1995); a
solid textbook, leads up to fractals
* Acheson D., "From calculus to chaos" (Oxford 1997)
(ISBN 0198502575); a very short and readable introduction to calculus, oscillations,
waves and chaos. Overlapping with PHYB54.
* Schroeder M., "Fractals, chaos, power laws. Minutes from an infinite
paradise." (W H Freeman and Co, 2000); lots of illustrations - not a textbook,
lots of nice digressions, great reading.
* Waldrop M., "Complexity", good additional reading for those extra 4
points (see below), as is
* Gleick, J., "Chaos: Making a New Science" (the classic, popular, intro
to science of chaos)
* Gradshteyn I.S. and Ryzhik, I.M. "Tables of Series, Products and Integrals" 1979
* Edward Lorenz, "The Essence of Chaos" 1999
* Haykin, "Adaptive filter_theory"
* Lowenstein, "When Genius Failed: The Rise and Fall of LTCM" 2001
* Ruelle, "The Mathematician Brain" 2007
* Puu "Attractors, Bifurcations, Chaos and Nonlin. Phenomena in Economics"
* Abraham and Ueda "Chaos: Avant-Garde Memoirs" 1993
* Wimberger,"Nonlinear
Dynamics and Quantum chaos" 2022 (2nd ed.), less recomm.
Cvitanovic et al, "Classical and Quantum Chaos" 2002
To the home page of Pawel Artymowicz
last modified: March 2024