- #1

- 14

- 0

## Homework Statement

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)

## Homework Equations

## The Attempt at a Solution

f

_{1}(x,y) = x(x-y)

x-nullcline: x(x-y) = 0 [itex]\Rightarrow[/itex] x = 0

f

_{2}(x,y) = y(2x-y)

y-nullcline: y(2x-y) = 0 [itex]\Rightarrow[/itex] y = 0

Fixed point: (0,0)

J(x,y) =

(2x-y -x )

(2y 2x-2y)

Therefore,

J(0,0) =

(0 0)

(0 0)

Thus,

0 = |J(0,0) - λI| = (-λ)(-λ) = λ

^{2}= 0 [itex]\Rightarrow[/itex] λ

_{1,2}= 0

___________________________________________________________________

Now this is where I get stuck; I have no idea where to go from here when λ

_{1}= λ

_{2}= 0.

I would really appreciate at least a little nudge in the right direction. Thank you.