MATLAB: Critical points for system of 3 first order differential equations

non-linear system

Hi all,
I am trying to solve a system of non-linear ODE's for the critical points. I then need to determine the stability of the points.
My system is as follows:
u'1 = u1 + 4*u2 – u1*u2
u2' = 9*u1 + 4*u2 – u2*u3
u3' = 2*u1^2 + 9*u2^2 -89
It was recommended to me to try Newton's method to find the critical points…but I am unsure how to do so for 3 equations.
The methods that I have learned up until this point won't work, considering the unknowns. I'm not asking for a solution…simply a nudge in the right direction.
Thank you

Best Answer

What would you define a critical point to be? Perhaps one where u'1=u'2=u'3=0?
How would you identify that set of points? Perhaps as solutions to the corresponding nonlinear system? It would seem the derivatives go away there. ;-)
Can Newton's method be applied to such a problem? Hint: do some reading about Newton-Raphson.