Three Critical Points and Type of Local Phaseportrait

[aeroguy2008]

Homework Statement

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?

 

[dirk_mec1]

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

 

[HallsofIvy]

Do you know the definition of "critical point"?

 

[aeroguy2008]

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

 

[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?

 

[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.

 

[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?

 

[The|M|onster]

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

 

[HallsofIvy]

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?

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.

 

[aeroguy2008]

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

 

[HallsofIvy]

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

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)!