A tiny part of the exponential fractal
A tiny part of the exponential fractal
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.
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%]
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
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. Cf. Solutions
Problem set 2, due on 24.02.24. See the 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.
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.
In this file, comments on the textbook can be found, which should help you
focus on things that may be seen in the exam. Remember that our textbook has
zilllions of problems to practice your knowledge, answers to odd-numbered
problems are in the companion book on our auxiliary page.
Please use both kinds in your preparation.
Here is an old exam with some quiz questions removed:
2014 exam pdf
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 $i_\infty =
..i^{i^{i^i}}$ really fast
⋄ Unofficial history of the nonlinear series transformation
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.
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)
Here are some 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 realistic.
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.