MATH 4700/6700 (Azoff)  Fall 2012
Qualitative Ordinary Differential Equations

This page was last updated on December 21.

Contents

  • Course Objective
  • Assignments
  • Exams
  • Project
  • Online Resources and Software
  • Course Syllabus
    •

    Course Objective

    Understanding and applying qualitative aspects of systems of ordinary differential equations.

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    Assignments


    #
    		 Due
    Read Section(s)
    Prepare for Class Discussion
    Hand In 
    		 Bonus/Grad
    0
    		 W 15 Aug
    		 1.0 thru 1.3


    1
    		 W 22 Aug
    		 2.1
    		 3, 4a
    		 1, 2
    2.2
    		 2, 4
    		 1, 3, 7, 8, 9, 10
    		 13abc
    2.3
    		 3b
    		 1a OR 1b, 2 (use a = k1 = k-1 = 1 for Part b)
    		 4
    2
    		 F 31 Aug
    		 2.4
    		 3, 5
    		 2, 4, 6, 7
    		 7, 9
    2.5 4
    		 1, 2
    		 3
    2.6 1, 2

    2.7 5, 6, 7
    		 1, 4
    2.8 1, 2a, 2c
    		 2(b), 3
    		 4, 7
    3
    		 W 12 Sept
    		 3.1
    		 2
    		 1, 3
    3.2
    		 1, 4
    		 2
    3.4
    		 2, 6, 7, 14
    		 4, 8, 13
    		 11
    3.5
    		 4, 7ab
    		 2*, 8
    3.6
    		 3, 5
    		 2
    3.7
    		 2, 6d
    		 3
    		 4
    4
    		 F 21 Sept
    		 4.1
    		 3, 7
    		 2, 5
    4.2
    		 2
    		 3
    4.3
    		 2, 5
    		 3, 6
    4.4
    		 2

    4.5
    		 3
    		1ab
    5.1
    		 1, 2, 8
    		 7, 10
    5.2
    		 4, 9, 11, 13ab
    		 3, 7, 10
    		 11, 14
    5.3
    		 2, 5

    5
    		 M 15 Oct
    		 6.1
    		 3, 9
    		 2, 6, 12a
    		 13
    6.2
    		1, 2
    6.3
    		 3, 13
    		 1, 8b, 11
    		 15
    6.4
    		 3, 5
    		 2, 4abd
    		 4e
    6
    		 M 29 Oct
    		 6.5
    		 4, 6, 9
    		 3, 12, 19abc
    		 8, 19d
    6.6
    		 1, 4a
    		 3, 4b, 5a, 7
    		 10
    6.7
    		 3
    		 1, 2abcd (worth 2 points)
    7
    		 F 9 Nov
    		 6.8
    		 1, 2, 9, 13
    		 4, 6, 7, 14
    		 11, 12
    7.1
    		 1, 3, 4
    		 2, 5, 8
    7.3
    		 1, 6b, 8, 9
    		 3, 10
    7.4
    		 1
    		 2
    Project Description
    		 some past projects listed below
    		 Include list of collaborators; use separate sheet of paper
    Bonus on Final
    (Cantor Set)
    		 11.1
    		 3, 4

    11.2
    		 4ab

    11.3
    		 1, 5, 8


    *For Problem 3.5.2,  the domain of  \phi is restricted to be in the interval (-pi, pi].  Otherwise, you do not need to know where Equation 3.5.7 on Page 66 comes from.  Note that Figure 3.5.6 occurs earlier, on Page 63.


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    Exams

    The comprehensive final exam was held from  Noon - 3 PM on Friday December 7.  The median score was 75%.

    There were review sessions starting at 1 PM on Wednesday December 5 and at 6 PM on Thursday December 6.

    Problems in BOLD typeface above could be used to help study for the test.

    The following topic guide was made available.

    Coverage for Final Exam

    Text Section
    		 Comment
    Chapter 1
    		 There are no problems here, but this is a good time to read this material to see how the course fits together.
    2.2
    		 You are not responsible for derivation of electrical circuit differential equation in Example 2.2.2 or for the mechanical analogue discussion at the end of Section 2.6
    2.4
    		 Centering Taylor polynomials at x* is an acceptable alternative to introduction of eta
    2.6
    		 Oscillations are most easily ruled out by the intermediate value theorem; you can ignore the mechanical analog subsection
    2.7
    		 Note that equilibrium points for dx/dt=f(x) are critical points for every corresponding potential V
    2.8
    		 Not responsible for improved Euler or Runge-Kutta methods on this test.
    3.1
    		 One should distinguish between bifurcation values and bifurcation points. 
    The tangential argument in Example 3.1.6 is less flexible than the necessary condition for bifurcation points that the partial derivative of f with respect to x be zero.
    3.3
    		 We did not cover this in class, but discussed a biological analogue involving diseases instead.
    3.5
    		 You are not responsible for this section. In particular, the legitimacy of ignoring the second derivative term is not adequately addressed until Section 6.3
    3.6
    		 You are not responsible for the bead on a tilted wire example, though we did set up the differential equation for the untilted wire in class, and later explained the legitimacy of ignoring the second derivative term in this case.
    3.7
    		 You do not need to memorize Equation (1) on Page 74.
    4.3
    		 You need only read through the fourth line on Page 98
    4.4
    		 It is good to read this now.
    4.5
    		 This will not be covered on the final.
    4.6
    		 You are not responsible for this
    5.2
    		 You are responsible for finding explicit solutions in cases of complex eigenvalues and when there is only one eigenvalue.
    5.3
    		 You are not responsible for this.
    6.3
    		 Pay close attention to the discussion of small non-linear terms on Page 151.
    Example 6.3.2 is easier to understand if you start with the polar form at the bottom of Page 153; it is the basic example of a linear center which is not a non-linear center.
    6.5
    		 Conserved quantities are often easily found by applying separation of variables to the equation for dy/dx obtained by dividing xdot by ydot.
    Pay close attention to Theorem 6.5.1
    6.6
    		 The system xdot=f(x,y), ydot=g(x,y) is also time reversible when f is odd in x and g is even in y.
    Pay close attention to Theorem 6.6.1.
    6.7
    		 You are not responsible for the cylindrical and tube diagrams.
    6.8
    		 We discussed this extensively in class.
    7.2
    		 We did not cover this.
    7.3
    		 Pay close attention to the Poincare-Bendixson Theorem.
    7.4
    		 Last section covered on the regular part of the final.


    There were two hour exams during the term. 

    The second hour exam covered through Section 7.4 of the text..  The median score was

    The first hour test, covered Chapters 2 through 5.  The median score, including correction opportunities, was

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    Project

    Project reports were be turned in at the final exam.

    Students formed small groups to choose projects on which to write reports and possibly give brief presentations. Project titles, brief descriptions, and lists of collaborators were due on Friday November 9.  The following were titles of submitted projects.

    Here are some project titles from earlier classes.

    Often our text follows up discussion of an example with a critique mentioning oversimplifications and providing references for more careful treatments.  See for example Section 3.7 on insect outbreaks, especially the concluding paragraph on Page 79.  Many projects could be based on following up such references.  In particular, Problem 7.1.9 provides such an opportunity.

    For the mathematically-minded, you might prefer giving an exposition of the proof of the fundamental existence and uniqueness theorem or explaining the role of the inverse function theorem in the study of bifurcations.

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    Online Resources and Software

    MIT lectures and applets:   http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/

    First order DE solver:   http://www.math-cs.gordon.edu/~senning/desolver/

    Second order DE solver:   http://www.zweigmedia.com/RealWorld/deSystemGrapher/func.html

    Comprehensive ODE software:  OdeFactory.com

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    MATH 4700/6700 (Azoff)
    Qualitative Ordinary Differential Equations
    Course Syllabus for Fall 2012

    Web Page  http://www.math.uga.edu/~azoff/courses/4700.html 



    Call Numbers
    		 43-117 for MATH 4700;  63-118 for MATH 6700



    Time & Place  1:25 - 2:15 MWF  303 Boyd



    Prerequisite
    		 Elementary Differential Equations (MATH 2700) and Linear Algebra (MATH 3000 or MATH 3500 or MATH 3500H)



    Objective Understanding and applying qualitative aspects of ordinary differential equations



    Text  S. T. Strogatz, Nonlinear Dynamics and Chaos, Perseus, ISBN#0738204536



    Topics Overview, Flows on the Line
    Bifurcations; Flows on the Circle
    Linear Systems; Phase Plane Analysis
    Limit Cycles and Bifurcations Revisited
    Lorenz Equations; One-Dimensional Maps
    Fractals and Strange Attractors
    		 About 2 weeks on each topic



    Grading Homework
    Hour Tests (2 @ 100 pts)
    Project 
    Final Exam    		 100 points 
    200 points
     50 points 
    150 points




    		Homework will be collected about once a week;  no late work will be accepted.  

    Students in 6700 are expected to hand in at least one grad/bonus problem on each assignment.

    The comprehensive final exam is scheduled for Noon - 3 PM on Friday December 7.



    Instructor  E. Azoff
         e-mail 
         Phone 
         Office     		 azoff@math.uga.edu
    542-2608 
    443 Boyd 
    Office Hours (Tentative)
    		 MWF:  8:30 - 9:30
    TuTh:    12:30 - 1:30
    except
    M-Tu Sept 17-18
    W       Sept 26
    M-Tu  Oct 1-2
    M-Tu  Oct 8-9

    First Assignments

    #
    		 Due
    Read Section(s)
    Prepare for Class Discussion
    Hand In 
    		 Bonus/Grad
    0
    		 W 15 Aug
    		 Chapter 1


    1
    		 W 22 Aug
    		 Section 2.1
    		 3, 4a
    		 1, 2
    Section 2.2
    		 2, 4
    		 1, 3, 7, 8, 9, 10
    		 13abc
    Section 2.3
    		 3
    		 1a OR 1b, 2 (use a = k1 = k-1 = 1 for Part b)
    		 4

    About Prerequisites

    The bulletin lists Introduction to Differential Equations (MATH 2700) and linear algebra (MATH 3000 or 3500 or 3500H) as prerequisites for 4700.  On the other hand, Strogatz (the author of our text) claims calculus knowledge as the only prerequisite to reading his book. The truth lies somewhat in between.  Prior exposure to linear algebra will be of use in dealing with linear systems of differential equations -- both for developing solution techniques based on eigenvalues and eigenvectors, and also for appreciating why linear approximation is the right way to go when dealing with the harder non-linear systems.

    Several members of our upcoming class did not take 2700 and I will try to strike a balance between accommodating them and not boring students who did take 2700.  You can get a good overview of 2700 from the homepage for the last time I taught the course, located at http://www.math.uga.edu/~azoff/courses/2700.html  

    A copy of the first hour test in 2700 can be found at http://www.math.uga.edu/~azoff/courses/2700t1.pdf .  A good goal is being able to solve the first three problems within the first few days of the start of this term.  In other words, everyone should feel reasonably comfortable with first order separable and linear differential equations and their applications; you can indeed read about these topics in most calculus books.  Almost all other 2700 topics are discussed in our text; as suggested above, we will review these briefly in class.

     A pdf version of the webwork assignments for MATH 2700 can be found at http://www.math.uga.edu/~azoff/courses/ww1108.pdf; the first five assignments correspond to Chapter 1 of Blanchard (the 2700 text).  You shouldn't feel that you have to solve all of these before the term starts, but it wouldn't hurt to look them over.

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