PHYD38 Nonlinear Systems and Chaos

PHYD38 Nonlinear Systems and Chaos

A tiny part of the exponential fractal

This page provides up to date syllabus (below), access to the lectures notes, assignments, and other material. Quercus does not have this information. It is used for announcements, and to submit & see the feedback regarding homework (in Assignments tab).

The 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 solve them when possible and/or linearize around the equilibrium points. This allows us to study the stability (or instability) of solutions. Starting from one-dimensional examples, we proceed to higher-dimensional cases, where new, interesting, behavior of dynamical systems emerges, including chaos. We draw on examples from physics and astrophysics, sometimes also biology and engineering. The course includes one research project performed 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, 3rd ed., 2024). Avoid the 1st and 2nd editions; there's not much wrong with them, but some topics and exercises are not identical. We don't want any confusion about page numbers. The 3rd edition is not much improved w.r.t. the 2nd, so maybe it would work, but that's your risk. 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, and an auxilliary 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 exam 37% = 19%(W) + 18%(Q)
project 16% = 8%(W) + 8%(P)
activity 3%
[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

Tutorials

Here is the page that elaborates on what we did in each tutorial. Useful for study before exams.

Absence at midterm and missed assignments

UTSC allows you to miss assignments and tests for one week per semester, because of illness or emergency. If you have such a need, write me an email, after self-declaring on Acorn site mentioned here. You can submit an AD up to 6 days before/after your absence. Attach your self-declaration file. I may be able to either allow a late submission of assignment (if solutions were not presented) or transfer your points to the final exam.

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. 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.

Problem set 0 (not graded)

[The 1st graded assignment will be due on 27 Jan. Please do the following exercise but do not submit it. You may volunteer to present your solution during a tutorial. We'll discuss the solutions then and perhaps try some other numerical methods on the problem. A similar method 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 is plagiarism, cf. below).
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 we will create during one of the tutorials.

Problem set 1, due on 27.01.24. See the A1 solutions
Problem set 2, due on 24.02.24. See the A2 solutions
Problem set 3, due on 17.03.24. See the A3 solutions
Problem set 4, Due on 3.04.24. See the A4 solutions

Note on plagiarism, including "AI" 😞

Plagiarism and cheating do happen, sometimes unwittingly. When I notice non-sequiturs in your written problem solution, close similarity of your solution to fellow student's or some web page, or close similarity to the output of generative AI, 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. A meeting in person may be called for that purpose (in accordance with UofT guidelines).
Looking up solutions on wiki, google, or using generative AI chatbots is not allowed in PHYD38, unless explicitly requested. One of the reasons is that those sources may have wrong answers to problems in physics. Another is that it's a plagiarism, and that it typically leads to an inability to solve problems in an exam (where you only have access to a standard scientific calculator w/o internet connection). Discussing the general methods for solving home assignments with fellow students is allowed, but the implementation, i.e. the written solution, must be (demonstrably) your own. And if you feel that you have to use ChatGPT or somebody's text from the web, you need to explicitly quote any such source.
The university-mandated procedures and/or penalties have to be used by lecturers against plagiarism. Read more in official docs.

The Julia set.

Midterm preparation

Midterm covers chapters 1 to section 6.4 (inclusive) of chapter 6 of Strogatz textbook -- everything that was discussed in Lectures (there are some older recordings in Media Gallery on Quercus, not exactly the same as this year). This means everything up to but excluding energy conservation and time-symmetry of 2-D systems.
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. Odd problems have solutions (cf. hidden sub-page). Those with odd numbers are solved in the back of the book, compare your solution to the solutions provided in kds.pdf.
Allowed aids: calculator, handwritten notes (4 pages on 2 or 4 sheets). Not allowed: Books, electronic devices, copies/printouts.

Here is the text of your midterm with solutions . This will (later) help you prepare for the final.

Preparation for the final on Tue 22 April 19:00-22:00 in BV264

Allowed and forbidden aids: same as midterm. The only difference: 8 pages of handwritten notes allowed (single or double-sided).

Comments gathered in this file, on what to review should help you focus on things that may be seen in the exam.
Here is a rather old exam (with some typos and some quiz questions removed): 2014 exam pdf
Old but perhaps a better training set, here is the 2017 exam (some Qs removed from quiz): 2017 pdf.
Notice, however, that we haven't done Henon's mapping from chapter 12, only Roessler map (the only topic required to know in our exam in 2025, from ch.12).
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, but no guarantee.

2025 exam

Here are final exam problems + solutions of Quiz. 2025 (pdf)

Preliminary results

See results

Quasi-blog: Variations on the theme of nonlinear dynamics

Here we discuss:
⋄ Simple numerical schemes
⋄ Physics is indeterministic
⋄ Theory of flight. Phugoidal oscillations in times of Zhukovsky, Lanchester and today
⋄ A dog, a duck and a puzzle of an invisible magnet
⋄ Perturbations or how to solve things we don't know how to solve
⋄ Analytical study of orbit subject to constant perturbing force (radiation pressure)
⋄ Roche lobe overflow as eigen-problem at L1 saddle point
⋄ Stability of Lagrange points in R3B
⋄ Simple numerical integrators: importance of the order for accuracy
⋄ Numerically solving $\ddot{s} = s(1-s^2) -\dot{s}^{\,q}$
⋄ 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
⋄ Turbulent jets
⋄ Proof of the superstability of Newton-Raphson's method
⋄ How to compute a new math constant $i_\infty = ..i^{i^{i^i}}$ faster
⋄ Unofficial history of the nonlinear series transformation

Student mini-projects

Presentations will take place on Monday 24 March

Group 1: Ming, Congyi, Kishan = Nonlinear filters, encryption Report
Group 2: Ningkun, Hongmiao, Yujun = Solitons Report
Group 3: Chad, Ali = Lyapunov and his work Report

Everybody gets the same grade, one-half of the mark is for the quality of the writeup, one-half for presentation. How you divide the 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 research after all.

Writeups

PDF (DOC only if you have to) are due 1 day before the presentation, as explained above. Please send me PDF by email. I'll post them & announce to you the location (URLs) via Quercus email, for a few days only. Everyone should read them before the presentation day and prepare 1 or more questions about the other projects.
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

Presentation

Student miniprojects will be presented in the form of slideshow 25 min long + 5 min discussion on 24 March. Don't forget to have a front page with the title and names of everybody, 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/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 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. Study solitons and their stability using analytical methods. Numerically compute propagation and interactions of soliton solutions to some equation.
10. Quantum chaos, e.g. based on S. Wimberger 'Nonlinear Dynamics and Quantum Chaos'
11. Your own idea (pre-approved by professor :-)

Additional topics

Supplementary to the Strogatz textbook. One or a few more may be presented, time allowing.
The 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.

I intend to mention topics denoted by ☆.

☆ More bifurcation diagrams of discrete mappings
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
☆ Nonlinear acceleration of the convergence of series
♣ Dynamics of galactic and protoplanetary disks
♣ Nonlinear waves, Fluid resonances, Particle resonances
♣ 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 are just one of the many 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
□ Ge-DoubleHopfbifurcation-2015.pdf Double Hopf bifurcation, 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)

Other recommended books

* Devaney R., "A first course in chaotic dynamical systems (...)" (1992) 2nd/3rd year math UTSG course 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
* 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 (classic popular book!)
* 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
* Hilborn, "Chaos, Nonlin. Dynamics 1985 (2nd ed.); good book, mentioned in our Etudes quasiblog.

To the home page of Pawel Artymowicz

_{last edit: Apr 2025}