Qualitative Differential Equations - Math 4700/6700

Instructor: Caner Kazancı
Offices: 525 Boyd Graduate Studies and 410 Driftmier Engineering Center, (706) 542 0863.
Office Hours: 10:45am-11:30am Tuesdays and and anytime by appointment.
Course Information: You can download the syllabus here.
Text Book: Nonlinear Dynamics and Chaos, by Steven H. Strogatz.
About this website: Please visit this website frequently to read important announcements (updates on test times, class schedule, syllabus etc.) and to get the homework assignments. Please notify me if you notice any inaccurate information.

Announcements

  • Collaboration Policy: You are encouraged to work together on homework assignments, but you should write your own solution. In other words, you are not allowed to see the actual paper that somebody else is handing in. This rule applies to computer codes and graphs as well. You may discuss formulas or algorithms with your friends; but you should write your own code. The only code you are allowed to see and modify is the one that is available through the course website. You should not send, receive, start as a template or look at your friend's code. Since your course grade is mostly based on HW grades, any penalty you receive will have a significant impact on your final grade. Therefore I strongly suggest that you pay extra attention to this matter.
  • Here's a short presentation about XPP.
  • Please e-mail me your groups.
  • Here is the main webpage of XPP. You can install XPP for Windows from here. It's free!
  • Detailed instructions for macs are included with the download. I suggest you follow these intructions. If you don't want to go through with that, you can try the following: Download the correct version of XPP for your computer from XPP website. Extract everything in a directory, say named "xpp". Open X11 (or XDarwin), and change your directory to "xpp". (eg. You can type "cd Desktop" to change your directory to Desktop. and you can type "ls" to list what files are in your current directory.) Place your ode code in the same directory "xpp". So, if you type "ls", you should see both "xppaut" and an ode file, say "competition.ode". Then type "./xppaut competition.ode". The XPP window should pop up.
  • You can use any basic text editor to work with ODE files. If you need an alternative, you can try "emacs" for Windows and "Aquamacs" for macintosh.
  • Here are some XPP codes: competition.ode, sir_epidemic.ode, bifurcation.ode
  • To print in XPP, go to "Graphic stuff -> Postscript". To print in Auto, go to "File -> Postscript". You will end up with a postscript file (with *.PS extension), which is similar to a PDF file. You can view and print this file as described here. If you have any issues, you can try capturing a screenshot of the plot instead.
  • Projects:
    Groups of 2-3 students will form around mid semester, and each group will work on a project. The project will consist of a presentation and a report. Projects will be presented on the final exam date, December 15, Tuesday between 8:00 and 11:00 am. Each group will have a 15-20 minute time slot, depending on the total number of projects. Reports are due the same date (in class).

    Project topics:
    You should form groups by October 5, and a project title must be determined by October 15. You can search for a project title based on your interest, using online resources, mathematical physics, chemistry, biology texts. I will go over some examples in class. It is important that you apply what you have learned in class on a problem, where your work provides (non-obvious, or non-trivial) insights that increase our understanding of the focus of your project.
    Presentation:
    Here's the outline of a typical presentation:

    1. Introduction to the field of study, physical problem description, clear description of research question and motivation.
    2. Transition to mathematical or computational problem description, including a clear description of each variable, parameter and equation. Reason why a specific method or model is preferred over others (if available).
    3. Presentation of what you have done once you have the mathematical model or equations, including methods, simulation and/or analysis (depending on the project).
    4. Discussion of results, insights gained (from your work above). Clear explanation of how the mathematical model helped us learn more about the system, that was not possible otherwise.
    Report:
    Reports will be similar to the presentation. The aim of the report is to provide a detailed description of your research project and results, which might not fit in a 15 or 20 minute presentation. Therefore the report should be more detailed, and mostly self contained (nothing essential should be referenced to other published work. However, citations to related work are welcome.). The report must be between 5-10 pages.
    Evaluation:
    I will not accept paper copies. I expect an e-mail from each group with all the following attachments before the due date above: 1. Presentation 2. Report 3. All codes related your presentation (if there are any). Any code that you have written must be included as a separate e-mail attachment, so I can regenerate the results my self.
    Report will form 10% of your final grade, and the presentation will form 20% of your final grade (5% for each four parts listed above).
  • Bold problem numbers are worth twice as regular ones.

Homework Assignments

  • Homework Set 1 (due Thursday in class, 8/27)
    Section 2.2 : 1., 7., 8., 9., 10.
    Section 2.3 : 2., 4.
  • Homework Set 2 (due Thursday in class, 9/3)
    Section 2.4 : 2., 4., 6., 7.
    Section 2.6 : 1.
    Section 2.7 : 1., 4.
    Section 2.8 : 2(b).
  • Homework Set 3 (due Thursday in class, 9/10)
    Section 3.1 : 1., 3.
    Section 3.2 : 2.
    Section 3.4 : 4,. 6., 8., 11.
  • Homework Set 4 (due Thursday in class, 9/17)
    Show your work on all problems, especially the ones in bold. Explain your answers in detail. Correct answers with no explanation will not get credit.
    Section 3.5 : 8.
    Section 3.6 : 2.
    Section 3.7 : 3., 4..
  • Homework Set 5 (due Thursday in class, 10/1)
    Section 5.1 : 7., 10.
    Section 5.2 : 1., 6., 8., 10.
  • Homework Set 6 (due Thursday in class, 10/8)
    Section 6.1 : 2., 3., 4., 6.
  • Homework Set 7 (due Thursday in class, or Monday by noon in Rebecca Gaff's mailbox at Boyd 4th floor.)
    (1.-4.) : Classify all fixed points using linearization for all problems in the previous HW above. (Counts as four regular problems.)
    (5.) : For the following system, find all the fixed points and classify them using linearization. Then sketch the null-clines, the vector field and a plausible phase portrait.
    x'=xy-1
    y'=x-y3
  • Homework Set 8 (due Thursday in class, 10/22)
    Section 6.3 : For the following problems, please draw the nullclines and the vector field as well: 1., 3., 4., 10(except d).
  • Homework Set 9 (due Thursday in class, 10/29)
    HW assignment
  • Homework Set 10 (due Thursday in class, 11/5)
    (1.-5.) : Do the following for the systems in problems 6.3.1., 6.3.2., 6.3.3., 6.3.4., 6.3.6.:
    • Use XPP to plot the scaled direction field. (Hit D, S, Enter)
    • Plot a couple of solutions (trajectories) to show the system around the fixed points. (Hit I, I, click on anywhere on the phase plane, hit Esc to stop)
    • Classify all the fixed points based on your observations (No need to do linear analysis.)
    • Draw the invariant sets of saddle nodes. (Hit S, M, click on a saddle node. Click on Yes, Yes, hit Esc several times. Don't worry about "out of bounds" errors.)
    Choose axis bounds carefully, so that all interesting behavior is displayed clearly. You won't need to adjust any parameters, default values work fine. All you need to include in your assignment for each system is a plot (that contains the direction field, some trajectories and invariant sets for saddle nodes) and a list of classification of each fixed point. You will need to do a screen shot, instead of a postscript output.
  • Homework Set 11 (due Thursday in class, 11/19)
    Section 6.5 : 4., 6.
    Section 6.8 : 3., 6., 8.
  • Homework Set 12 (due Thursday in class, 12/03)
    Section 7.1 : 2., 3.
    Section 7.2 : 7., 12., 13.
    Section 7.3 : 1.
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