MA508- Homework 4 Solved

You will complete each of twelve homework assignment as part of a three- or four-person group. Group members are assigned randomly from the class and will remain the same for the duration of the quarter. Each group turns in one homework, and each participating group member receives the same grade on the assignment. One member of the group is responsible for writing the homework (the writer), and this writer rotates for every assignment.

Homework groups work best if: Each member of the homework group finishes (or honestly attempts) the homework independently. At some appointed time, well before the due date, the group meets and everyone compares answers. Any discrepancies are discussed until a consensus is achieved. The writer notes the group consensus and makes sure she or he understands how to do the problem. After the meeting, but before class, the writer neatly and clearly writes the homework according to the Homework guidelines.

Homework groups don’t work if: One or more of the members skips meetings; each group member does not honestly attempt the homework prior to the meeting; a consensus in not reached for each assigned problem. If a group member does not adequately participate in the homework, write a note on the homework and alert your PLA. That person will not receive credit.

Homework guidelines for writers:

(Adapted from the website of Professor Andy Ruina). To get full credit, please do these things on each homework.

As a group writer, upload your homework as a single PDF on the day it is due.At my discretion, late homework may or may not be accepted for reduced credit.
On the first page of your homework, please do the following. On the top left corner, please put thecourse information, homework number and date, e.g.:
M

On the top right corner, please put your group number, the names of your group members, with the writer at the top and clearly indicated. Also indicate any non-participating group members, e.g.:

Group 3

Sidney Crosby (writer)

Marissa Tomei

Michelle Wie

Barry Obama (did not participate)

CITE YOUR HELP. At the top of each problem, clearly acknowledge all help you got from faculty, students or any other source (with exceptions for lecture and the text, which need not be cited). You could write, for example: “Mary Jones pointed out to me that I had forgotten to divide by three in problem 2,” or “Nadia Chow showed me how to do problem 3 from start to finish,” or “I copied this solution word for word from Jane Lewenstein” or “I found a problem just like this one, number 9, at cheatonyourhomework.com, and copied it,” etc. You will not lose credit for getting and citing such help. Don’t violate academic integrity rules: be clear about which parts of your presentation you did not do on your own. Violations of this policy are violations of the WPI Code of Academic Conduct.
Your work should be laid out neatly enough to be read by someone who does not know how to do theproblem. For most jobs, it is not sufficient to know how to do a problem, you must convince others that you know how to do it. Your job on the homework is to practice this.
This homework covers 1. Classification of fixed points (2+D); 2. Linearization (2+D); 3. 2D phase portraits near a fixed point.

These topic are covered in §5.0-6.3 in Strogatz.

Part of getting good at drawing phase portraits is experience. Here are four relatively simple systems (all from the book) to help you practice.

For each of the following (problems 1-3):

Find all fixed points and classify them;
Sketch the local phase portrait near each fixed point.
1.

x˙ = 1 + y−e−x y˙ = x3−y

2.

x˙   = xy− 1 y˙              = x−y3

3.

x˙         = x(3 −x−y) y˙      = y(2 −x−y)

Turn in a completed version of worksheet 4, which you worked on during class