Math
5470/6440 Chaos and Nonlinear Systems
T Th 9:10 am  10:30 am LS 101

Text: Nonlinear Dynamics and Chaos
by Steven Strogatz
Instructor: Prof. Elena
Cherkaev
Office: LCB 206 ph: 5817315
email: elena@math.utah.edu (please write 5470 in the subject line)
Office hours: T 10:3011:30 am and by appointment

Class
webpage: http://www.math.utah.edu/~elena/M5470/5470.html
Midterm 1  March 1: Ch 25
Midterm 2  April 12: Ch 58
Final  Friday, April 27, 2018, 8:00 – 10:00 am
*** A twosided formula sheet is allowed on the final
Review session: April 25, 2:00 pm  2:50 pm, LCB 218
To
receive full credit please
show all
work!
Steven Strogatz on synchronization
Explore fractals at http://users.math.yale.edu/public_html/People/frame/Fractals/

Assigned Homework:

Boldfaced problems = "must solve", the rest is recommended
Jan 11: Ex 2.2 # 3, 4, 6, 7, 8, 10, 12; Ex 2.3 2, 4, 5, 6
Jan 16: Ex 2.4 # 1, 2, 3, 4, 5
Ex 2.6 # 1
Jan 18: Ex 3.1 # 1, 3; Ex. 3.2 # 2, 4
Jan 23: Ex 3.4 # 3, 5, 11
Jan 25: Ex 3.4 # 8, 14, 15, 16
Jan 30: Ex 2.7 # 1, 3, 4, 6 Ex 3.5 # 7, 8
Feb 1: Ex 3.6 # 6, 7;
Ex 3.7
#
5, 6 ;

Feb 8: Ex 4.1 # 3, 4, 7, 8; Ex 4.2 # 1, 2, 3;
Feb 13: Ex 4.3 # 1, 3 ; Ex 4.4 # 2, 3; Ex 4.5 # 1, 3;
Feb 15: Ex 5.1 # 1, 2, 9, 10, 11
Feb 20: Ex 5.2 # 1, 3
Feb 22: Ex 5.2 # 7,12 ,13,14;
Ex
5.3
# 26;

Feb 27: Ex 6.1 # 1, 3, 5; 6.1 # 811;
Mar 1: Midterm 1
Mar 6: Ex 6.2 # 2; Ex 6.3
# 1, 2, 3, 12, 13,
14, 15 ;
Mar 8: Ex 6.4 # 1, 3; Ex 6.8 # 6,
7, 8 , 9, 11, 13;
Mar 13: Ex 6.5
# 1, 2, 8, 9, 11
Mar 15: Ex 7.1 # 15, 7; Ex 7.2 #
13, 6, 9; Ex 7.3 # 1, 7;

Mar 1825: Spring Break

Mar 27: Ex 8.1 # 14, 5, 6; Ex
8.2 # 1, 2; Ex
8.4 # 1, 2
Mar 29: Ex
8.6 # 1, 3, 7; Ex 8.7 # 9;
Apr 3: Ex 9.3 # 27, 9, 10. Numerically simulate Lorenz attractor for
different parameters of the system, see example in maple: lorenzmaple.html
Apr 5: Ex 10.1
# 10, 11, 12, 13; Ex 10.3 # 1, 2,
4 , 7, 8, 9;
Apr 12: Midterm 2
Steven Strogatz, Universal aspects of period doubling:
https://www.youtube.com/watch?v=ol6aQcgohxI&list=PLbN57C5Zdl6j_qJApARJnKsmROzPnO9V&index=20&t=0s
Steven Strogatz, Feigenbaum's renormalization analysis of period doubling:
https://www.youtube.com/watch?v=9OkVNInimSc&list=PLbN57C5Zdl6j_qJApARJnKsmROzPnO9V&index=21&t=0s
Apr 17: Ex 11.1 # 5, 6; Ex 11.2 # 1, 2, 5, 6
Apr 19: Ex 11.3 # 1, 3, 4, 5, 7, 8, 9; Ex 11.4 # 1, 2, 3

Project is due on May 10 (please talk to me if you need more time).
The
project is a short paper, numerical project, or other original creative
work on any topic of dynamical systems, chaos, or fractal geometry
related to the course. If the project deals with modeling of some
physical or biological phenomenon, then it should include description of
the problem, formulation and justification of the mathematical model,
analysis of the model, analytical and/or numerical results. It might
include a series of models improved in various aspects, references, the
project might be a review of research paper. The project on iterated
maps, fractals, and strange attractors, should also have description and
formulation of the problem, definitions, exposition of the used methods
and techniques, analytical and numerical results of investigation,
references for the used materials. If your project has numerical
simulations, please submit the code.
Ideas for project topics can be found in the textbook among the problems
we have not discussed in detail. One can begin with the text problem
and follow up on the references. Or one can choose an interesting paper,
write a report on it and try to add some analysis or numerical
simulation.
Some topics for possible projects:
Pattern formation in fluid systems: RaleighBernard convection, Couette flow
Ising model of magnetism
Zebra stripes and butterfly wing patterns: A biochemical switch
Josephson Junction array dynamics
Computation of orbit diagram and Liapunov exponent
Investigation of decimal/binary shift map
Sarkovski's theorem
Constructing fractals as fixed points of iterated maps
Fractal dimension of cracked surface
Computation of correlation dimension
There are various interesting projects related to strange attractors: look at Rossler system,
Baker's map (12.1.36), Henon's map (12.1.8, 12.2.112), Doublewell oscillator (12.5.15), etc.
Here I put several papers that can be used for a project. Other resources can be used for a project as well.
For instance, a recent issue of SIAM news deals with nonlinear dynamical systems:
https://sinews.siam.org/CurrentIssue/IssueArchives/IssueArchivesListView/PID/2282/mcat/2279/evl/0/TagID/267?TagName=Volume51Number2March2018
Here is a couple of articles from that issue:
Matthew R. Francis, Selforganization in Space and Time
https://sinews.siam.org/DetailsPage/selforganizationinspaceandtime
Daniel J. Gauthier, Reservoir Computing: Harnessing a Universal Dynamical System
https://sinews.siam.org/DetailsPage/reservoircomputingharnessingauniversaldynamicalsystem
The list of possible projects just
gives examples. If you want to work on your own topic, you are welcome
to
do so. In this case please discuss the topic with me. Feel free to
explore, investigate, and apply what we studied.
The project is due: May, 10.
By submitting this assignment, you are representing that it is your own work and that you have followed the rules associated with the assignment. Incidents
of academic misconduct (including cheating, plagiarizing, research
misconduct, misrepresenting ones own work, and/or inappropriately
collaborating on an assignment) will be dealt with severely, in
accordance with the Student Code (http://www.regulations.utah.edu/academics/6400.html).
Academic misconduct may result in a failing grade for the course and
even more severe measures.

Course description Chaos
is everywhere around us from fluid flow and the weather forecast
to the stock exchange and striking geometric images. The theory
of nonlinear dynamical systems uses bifurcations, attractors and
fractals to describe the chaotic behavior of real world things.
The course gives an introduction to chaotic motions, strange
attractors, fractal geometry. The emphasis of the course is on
applications:
 Mechanical vibrations
 Chemical oscillators
 Superconducting circuits
 Insects outbreaks
 Genetic control systems
 Chaotic waterwheels
 Chaotic communications


The course is addressed to senior undergraduate and
graduate students in mathematics, science and engineering.
Prerequisites: Calculus and Differential Equations.
Tentative Course Schedule:
Part I: Jan 8 
Jan 25  Flows on the line. Bifurcations. Flows on the circle.
Part II: Jan 30 
Mar 6  Linear systems: Phase plane. Limit Cycles. Bifurcations.
Part III: Mar 6
 Apr 24  Lorentz equations. 1D
Iterated maps. Fractals. Strange attractors.
Exams: There will be two midterms (100 pts each), final (200 pts), and an optional
project. Tentative dates for the
midterms: Feb 22; Apr 12.
Final test: Friday, April 27, 2018, 8:00 – 10:00 am
Grading: The grade will be
calculated as an average of the midterms and the project and/or
final.
Holidays: Martin Luther King Jr.
Day: Monday, January 15; Presidents'
Day: Monday, February 19
Spring break:
March 1825
Computer
lab set up: Maple
example
(slope field)
Maple: Lorenz attractor
For Matlab, download dfield8 and pplane8 from John C.
Polking's website: http://math.rice.edu/~dfield/index.html#8.0
Java simulator: http://chaos.wlu.edu/106/programs/lorenzdes.html
ADA statement: The American with
Disabilities Act requires that reasonable accommodations be
provided for students with physical, systemic,
learning, and other disabilities. Please contact me at
the beginning of the semester to discuss any such
accommodations for the course.