# Help me learn to sketch phase portraits!

1. Jan 16, 2012

### Ribbed

Hey guys,

Been struggling with this all day. I've got the following situation:

x1*=-x1+x2

x2*=-2x1-x2

Where the 1&2 after the x are subscripts.

Now i can find the equilibrium no trouble: (0,0). I also know, by looking at the eigenvalues that it is a sink.

But how on earth do you sketch it? I've found nullclines:

-x1+x2 = 0

-2x1-x2 = 0

But i cannot, for the life of me, tell which way these lines are going.

Help really appreciated.

2. Jan 16, 2012

### wisvuze

You want to bring it into a cannonical form, this system will be a spiral sink with eigenvalues - 1 +/- sqrt( 2 i ) so it will be a spiral sink. The cannonical form for a case like this, is a matrix like:
a -b
b a

The lines will go inward since it is a sink, and if a > 0 then the spiral goes counterclockwise

3. Jan 16, 2012

### Ribbed

Thanks for your reply. What i posted above is just a preliminary example that i cannot do, i've asked my professor too.

I've never seen ( i dont think) the method you have mentioned?

Are there any other ways? I had in mind something like:

dy/dx=f(x,y)/g(x,y)

(Of course, where my x and y are x1 and x2, and i could look at any point x1,x2 and see its direction).

I really need quite a bit of intuition in this as i can not see what is going on.