University of Utah
Department of Mathematics

Math 5470/6440   Nonlinear Dynamics and Chaos
M W F     2:00-2:50 pm     LCB 225 

Text:  Nonlinear Dynamics and Chaos 
           by  Steven Strogatz

Instructor:   Prof. Elena Cherkaev
Office: LCB 206   ph: 581-7315   email: elena@math.utah.edu
Office hours: W 3-4pm and by appointment


Test:   Wednesday March 16 -- Sample problems for practice test: 3.4.6, 3.4.16, 6.3.4, 7.1.2, 7.2.6(a)

Final test:  April 29,  1:00 -- 3:00 pm       Sample test (last year final)

Syllabus

Maple/Matlab webpage
 Lorenz attractor    --    Java simulator:     http://chaos.wlu.edu/106/programs/lorenzdes.html


List of possible projects

The project is not just one of the exercises from the book. 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 numerical results. It might
include a series of models improved in various aspects, reference(s) and review of research paper(s).
The project on iterated maps, fractals, and strange attractors, should also have description and formulation
of the problem, required definitions, exposition of the used methods and techniques, analytical and numerical
results of investigation, reference(s) for the used materials and review of research paper(s).

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 during the finals week, on or before May 6.    

To receive full credit please show all work! 

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Assigned Homework Problems:    ( Please,   NO LATE HOMEWORKS! )
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Hw 1 :  Due   Jan 25
Jan 12 :      Ex 2.2  #  1,  3,  4,  7,   8;    Extra credit:  # 6, 12    
Jan 14:      Ex 2.3   # 2,  3,  4 ;    Ex 2.4  #  5,  6
Jan 19:      Ex 2.7  #  1,  4,  6
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Hw 2:   Due   Feb 9
Jan 21 :      Ex  3.1 #  1*,  3* ; 
Jan 24 :      Ex  3.2 #  2*,  5*;   Extra credit:  3.3.1,  3.3.2
Jan 26 :      Ex   3.4 #  3*,  8*,  14*,  16*    Extra credit:  3.4.15
Jan 28 :      Extra credit:    #  3.6.6,   3.6.7 ,   3.7.5,   3.7.6
Jan 31 :      Ex  4.2  #2;   Ex  4.3   #  1*,  2,   3* ;  Ex  4.5  # 3*
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Hw 3:   Due   Feb 23
Feb 2 :      Ex  5.1 #  2* ; 
Feb 4 :      Ex  5.2  #  2,  3*,  7*,  9*,  12
Feb 7 :      Ex  5.3   #  2 - 6
Feb 11 :     Ex  6.1   # 1*,  4*,  6*   Ex  6.2.2*    Extra credit:  6.1.8 - 6.1.11
Feb 16 :     Ex  6.3   # 2,  3*,    13,  14*    Extra credit:  6.3.4,  6.3.15
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Hw 4:   Due   March  9
Feb 16 :     Ex  6.4  # 1,   Ex  6.5   # 2*,  8*,  9*
Feb 18 :     Ex  6.8  #  6,  7*,  8*   Extra credit:   11,  12
Feb 23 :     Ex  7.1  #  2, 3, 5*;  Ex  7.2  # 3*, 6(a)*, 12;  Extra credit:  7.1.8
Feb 28 :     Ex  #  7.3.9 ;  7.5.4* ;  Extra credit:   7.3.10
Mar 4 :      Ex  8.1  #  2,  3*, 4*;  Extra credit:   8.1.6,  8.1.11
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Homework is individual work. You are encouraged to discuss homework problems with friends and make study groups, but each homework should be written individually. Identical homeworks will be punished. To receive credit please show your own work!
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Hw 5:   Due   March  30
  Mar 7 :     Ex  8.2   #  1*, 2, 9(a-c) ; Extra credit:   8.2  # 3, 5, 9(d)
  Mar 11 :   Ex  8.6  # 1,  2*(a-c) ;   Extra credit:   8.6.3
  Mar 14 :   Ex  8.7  # 1*, 2,  3* ,  9*
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Mar 21 - 26:      ----------------   Spring Break  ----------------
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Hw 6:   Due   April  13
  Mar 28 :  Ex  9.2  # 3, 6 ;  Ex  9.4.2*;  Extra credit:  9.3  # 9, 10
  Mar 30 :  Ex  9.3  # 1*, 8* ;   Extra credit:  9.5  # 1-4
  Apr  1 :    Ex  10.1  # 11* ;  Ex  10.3  # 1*
  Apr  6 :   Ex  10.3  # 2* , 4*;  Ex  10.4  # 3* ; Ex  10.5  # 2
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Hw 7:   Due   April  27
  Apr 11 :    Ex  10.3  # 7*, 8*, 9;  Ex. 11.1 # 4-7*  
  Apr 15 :    Ex. 11.2  # 4*,  6* ;   Ex. 11.3  # 1*,  2*, 5*
  Apr 18 :    Ex. 11.4  # 1*,  6*,  9
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* - Starred numbers will be graded

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 Outline

Part I
Jan 10-
  Feb 2

 Flows on the line
 Bifurcations
 Flows on the circle

 Jan 10-17
 Jan 19-28
 Feb 1 - 2

Part II 
Feb 4-
  Mar 9

 Linear systems
 Phase plane
 Limit Cycles
 Bifurcations

 Feb 4 - 7
 Feb 9 - 18
 Feb  21 - 25
 Mar 2 - 9

Part III 
Mar 11-
   Apr 27

 Lorentz equations
 1D Iterated maps
 Fractals
 Strange attractors

 Mar 11 - 18
 Mar 28-Apr 6
 Apr  8 - 18
 Apr 20 - 27

Apr. 29

 Final exam

   1:00 – 3:00 pm

                                                                                           Holidays:

Martin Luther King Jr. Day

Monday, January 17

Presidents’ Day

Monday, February 21

Spring break

March 21-26

Exams: There will be one midterm and a final. Tentative date for the midterm: Friday, March 11. Probable course material is chapters 1-7.

Grading:  Your percentage grade will be calculated as follows: 40% for the homeworks; 20% for the project, for one midterm, and for the final.

Homework: Homework will be collected biweekly on Mondays, and a portion of the problems will be graded. You are encouraged to discuss homework problems with friends and make study groups, but each homework should be written individually. (Copying someone else's work will be punished).


ADA statement:  The American with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, and learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations for the course.