# Homework Help: Three Critical Points and Type of Local Phaseportrait

1. Jun 26, 2008

### aeroguy2008

1. The problem statement, all variables and given/known data

y1'= −4*y1+y2+y1*y2
y2'= −2*y1−y2+y1*y1

Determine the three critical points of the system and their type of local phase portrait (stable node, unstable, saddle point, spiral, center, no node)

Hence I need to get three critical points (x1,y1), (x2,y2) & (x3,y3) and their local phase portraits. Can somebody help pls?

2. Jun 26, 2008

### dirk_mec1

Set both y1' and y2' to zero and solve to get three solutions.

3. Jun 26, 2008

### HallsofIvy

Do you know the definition of "critical point"?

4. Jun 28, 2008

### aeroguy2008

Is a critical point when I differentiate the function and put it to zero?

5. Jun 28, 2008

### HallsofIvy

You already have the derivatives!

As dirk mec1 said, set each of those derivatives equal to 0 and solve for y1 and y2. (The way your system is set up, each critical point will be of the form (y1, y2).)

Now, how do you determine the phase portrait type?

6. Jun 28, 2008

### The|M|onster

To elaborate even further, a y1' = 0 and y2' = 0 are your nullclines, and the critical points can be thought of as the intersections of these nullclines.

Last edited: Jun 28, 2008
7. Jun 28, 2008

### aeroguy2008

Thanks....once I set these two equations I got y values of; y1,1=0, y1,2=-1, y1,3=2...how do i move on from here?

Last edited: Jun 28, 2008
8. Jun 28, 2008

### The|M|onster

Well, what are the y2 values that correspond to each of your y1 values?

9. Jun 28, 2008

### HallsofIvy

Do you mean that the three values of y1 are 0, -1, and 2? What is the corresponding value of y2 for each y1?

I'm a bit concerned about the whole tenor of this thread. If you are at a point in a course where you are expected to be able to draw local phase portraits, or determine whether a given critical point is a node, center, etc., finding the points themselves should be trivial. Yet you sound like you really have no idea what the problem is asking.

10. Jun 28, 2008

### aeroguy2008

Well you can be a whole lot concerned. I am just trying to learn something.

11. Jun 29, 2008

### HallsofIvy

Then I can be a lot less concerned. I was afraid you were taking a course in differential equation (and might have to take the final exam next week)!

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