Find the fixed points and classify them using linear analysis. Then sketch the nullclines, the vector field, and a plausible phase portrait.
dx/dt = x(x-y), dy/dt = y(2x-y)
The Attempt at a Solution
f1(x,y) = x(x-y)
x-nullcline: x(x-y) = 0 [itex]\Rightarrow[/itex] x = 0
f2(x,y) = y(2x-y)
y-nullcline: y(2x-y) = 0 [itex]\Rightarrow[/itex] y = 0
Fixed point: (0,0)
(2x-y -x )
0 = |J(0,0) - λI| = (-λ)(-λ) = λ2 = 0 [itex]\Rightarrow[/itex] λ1,2 = 0
Now this is where I get stuck; I have no idea where to go from here when λ1 = λ2 = 0.
I would really appreciate at least a little nudge in the right direction. Thank you.