system differential equations

dirk_mec1

1. The problem statement, all variables and given/known data

Determine the equibrilium solutions and their stability properties of the system below:


[tex]
\dot{x} = (1-z)[(4-z^2)(x^2+y^2-2x+y)-4(-2x+y)-4]
[/tex]

[tex]
\dot{y} = (1-z)[(4-z^2)(xy-x-zy)-4(-x-zy)-2z]
[/tex]

[tex]
\dot{z} = z^2(4-z^2)(x^2+y^2)
[/tex]



3. The attempt at a solution
The critical point (0,0,1) is difficult to characterise since the eigenvalues are 0, 0, 0. However you can determine stabilitity of this point by looking at the flow of z' ... but how?