- #1

ceejay2000

- 6

- 0

## Homework Statement

http://www.freeimagehosting.net/t/9369y.jpg

## Homework Equations

## The Attempt at a Solution

a) is as follows: http://www.freeimagehosting.net/t/4oqft.jpg

Then for b), I have the equilibria as [itex] (0,0,0) [/itex] and [itex] (r-1,\frac{r-1}{r},\frac{(r-1)^2}{r}) [/itex]

To examine their stability I have the Jacobian;

[itex]J=\begin{vmatrix}

1-\lambda& r& 0\\

1& -1-\lambda& -1\\

y& x& -1-\lambda

\end{vmatrix}[/itex]

giving [itex] J_{(0,0,0)}=(-1-\lambda)[(-1-\lambda)^2 -r) = 0 [/itex]

So, [itex]\lambda^2 +2\lambda +(1-r) =0[/itex] and the stationary point is stable for [itex]r<1[/itex] by Routh-Hurwitz.

Next, [itex]J_{(r-1,\frac{r-1}{r},\frac{(r-1)^2}{r})}=1-r-2\lambda-3\lambda^2-\lambda^3= 0[/itex]

and by Routh-Hurwitz:

[itex]r<1[/itex] again

and for the [itex]p_1p_2>p_3[/itex] part, we have:

[itex]6>r-1 \Rightarrow r<7[/itex]

which it is anyway from [itex]r<1[/itex]

For the Hopf Bifurcation at the nontrivial stationary point, I took [itex]\lambda= i w[/itex] and equated the imaginary and real parts to then give [itex]r=7[/itex]

After this I'm not sure what is meant by

*"What is the frequency of small disturbances at the point of Hopf bifurcation?"*and also have never been very good at sketches haha so any help here would be appreciated.

I got stuck with part c) also so I'm not entirely sure whether the above is correct or not. I thought to linearise the system about this point I just had to use the Jacobian from above and input the nontrivial solutions and the value of [itex]r[/itex] at the Hopf point but this didn't yield any zero eigenvalues which I thought was necessary for the center manifold?

Maybe I've got confused somewhere here so again any help would be most appreciated!

**Thanks!**