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: 581-7315  
   email:   elena@math.utah.edu
  (please write  5470  in the subject line)
   Office hours: T 10:30-11:30 am and by appointment

  Class webpage:  http://www.math.utah.edu/~elena/M5470/5470.html

Midterm 1 - March 1: Ch 2-5
Midterm 2 - April 12: Ch 5-8
Final - Friday, April 27, 2018,  8:00 – 10:00 am 

*** A two-sided 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/

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  Assigned Homework:  
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Bold-faced problems = "must solve", the rest is recommended
Jan 11:  
Ex 2.2  #  34,  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  #  
814, 15, 16
Jan 30:   Ex 2.7  #   1,  3,  46    Ex 3.5  #  7, 8
Feb 1:   Ex 3.6  #  6, 7;   Ex 3.7  #   5,  6 ;  
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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  # 2-6; 
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Feb 27:   Ex 6.1 # 1, 3, 5
6.1 # 8-11
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   #  1-5, 7Ex  7.2  #  1-3, 6, 9;   Ex  7.3 # 1, 7;
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Mar 18-25:  Spring Break
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Mar 27:   Ex  8.1  # 1-4, 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 # 2-7, 9, 10.   Numerically simulate Lorenz attractor for
  different parameters of the system,
see example in maple: 
lorenz-maple.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_qJA-pARJnKsmROzPnO9V&index=20&t=0s

Steven Strogatz, Feigenbaum's renormalization analysis of period doubling:
https://www.youtube.com/watch?v=9OkVNInimSc&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&index=21&t=0s

Apr 17:     Ex 11.1 # 5,  6;   Ex 11.2 # 1, 2, 5,  6    
Apr 19:     Ex 11.31, 3, 4, 5, 7, 8, 9Ex 11.41, 2, 3     
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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: Raleigh-Bernard 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 (12.1.3-6), Henon's map (12.1.8, 12.2.1-12), Double-well oscillator (12.5.1-5), 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/Current-Issue/Issue-Archives/Issue-Archives-ListView/PID/2282/mcat/2279/evl/0/TagID/267?TagName=Volume-51-|-Number-2-|-March-2018

Here is a couple of articles from that issue:
Matthew R. Francis, Self-organization in Space and Time
https://sinews.siam.org/Details-Page/self-organization-in-space-and-time
Daniel J. Gauthier, Reservoir Computing: Harnessing a Universal Dynamical System
https://sinews.siam.org/Details-Page/reservoir-computing-harnessing-a-universal-dynamical-system

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/6-400.html). Academic misconduct may result in a failing grade for the course and even more severe measures. 
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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 18-25


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.