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.