Hopf Bifurcation - Nonlinear Diff Eqns

[ceejay2000]

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 $(0,0,0)$ and $(r-1,\frac{r-1}{r},\frac{(r-1)^2}{r})$


To examine their stability I have the Jacobian;


$J=\begin{vmatrix} 1-\lambda& r& 0\\ 1& -1-\lambda& -1\\ y& x& -1-\lambda \end{vmatrix}$


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


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


Next, $J_{(r-1,\frac{r-1}{r},\frac{(r-1)^2}{r})}=1-r-2\lambda-3\lambda^2-\lambda^3= 0$

and by Routh-Hurwitz:

$r<1$ again

and for the $p_1p_2>p_3$ part, we have:

$6>r-1 \Rightarrow r<7$

which it is anyway from $r<1$


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


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 $r$ 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!

 

[ceejay2000]

Bump

Anyone got any ideas about this? Still haven't figured out if/where I might have gone wrong!