Drawing phase portrait

[MaxManus]

Homework Statement

I want to draw a phase portrait from linearization of a nonlinear differential equation with eigenvalues l1 =1 and l2= -1.
With equilibrium in (2,2)

x' = y-x
y' = (x-1)(y-2)

Homework Equations

The Attempt at a Solution

First I compute the eigenvectors v1 = [0;1] v2 = [1;2]
I then draw v1 as a horizontal line through the point (2,2).
I then check the direction of x' to left and to the right of (2,2). F.ex x = 0 y = 2, then x' = 2. x = 3, y = 2, then x' = -1. So the direction on both sides are against the (2,2)

Then I draw v2. I draw a straight line through (2,2) and (3,4). and check the direction to the left and to the right of (2,2). x = 1, y = 0, then x' = -1 and y' = 0, something wrong hear?, x = 3, y = 4, x' = 1 and y = 4. So the direction on both sides are away from (2,2)

Is this the right idea for drawing phase portraits?

Edit: There are two equilibrium points. The second one is a spiral sink in (1,1) which goes clockwise

 

[lippyka]

. To draw the phase portrait, I will draw the eigenvectors v1 and v2 through the equilibrium point (2,2). v1 = [0;1] and v2 = [1;2]. I will draw v1 as a horizontal line through the point (2,2). I will then check the direction of x' to left and to the right of (2,2). F.ex x = 0 y = 2, then x' = 2. x = 3, y = 2, then x' = -1. So the direction on both sides are against the (2,2)Then I draw v2. I draw a straight line through (2,2) and (3,4). and check the direction to the left and to the right of (2,2). x = 1, y = 0, then x' = -1 and y' = 0, something wrong hear?, x = 3, y = 4, x' = 1 and y = 4. So the direction on both sides are away from (2,2)I then draw the spiral sink at the (1,1). I will draw a circular arc centered at (1,1) and going clockwise. I will then check the direction of x' to left and to the right of (1,1). For example, x = 0 y = 1, x' = 1 and y' = 0. x = 2, y = 1, then x' = -1 and y' = 0. So the direction on both sides are away from (1,1). Finally, I draw the trajectory between the two equilibriums, connecting the two eigenvectors. This is the correct idea for drawing a phase portrait.