# How to find equilibrium points of a system of 5 non linear ordinary differential equations???

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Rose on 21 Nov 2013
Edited: Alan Weiss on 22 Nov 2013
Hello all,
I have a system of 5 non linear ordinary differential equations with variable coefficients (with at least 3 parameters that are unknown and rest of them are known). I am trying to find the equilibrium points by hand but it seems like it is not possible without the help of a numerical method. What would be a good method to calculate equilibrium points of the system? (I saw thousands of examples on internet but they use systems of two dimensional ODEs with constant coefficients, which seems to be the 'easiest' case..)
Another question (somehow related to the problem above): Would it be possible to check the stability of the equilibrium points and then draw a bifurcation diagram? If so, please suggest some way out!
Thank you very much for taking time out of your busy schedule to read/ answer my question! Really appreciated!
Marc on 21 Nov 2013
"equilibrium points"??? Maybe it is syntax but ODES change as a function of something. Change as a function of time, space, etc. etc. so what do you mean that you are trying to find 3 parameters at equilibrium? Or are you trying to find 3 parameters where dX = 0 and X is a vector with five variables (5 non linear ODES)... Can you show us your system? If you set dX to 0, do you end up with a system of non-linear equations? Can you then solve for those parameters? Although with the way you described this, then you end up with an over determined set, so SVD or some optimization algorithm (fminsearch, fmincon, fminunc.....)?????? Good luck but I think we will need a lot more information to be of any help.

Alan Weiss on 22 Nov 2013
Edited: Alan Weiss on 22 Nov 2013
Basically you want to find a point where the derivative of each equation is zero. I mean, if your equations are
d/dt x(t) = F(x), where x and F are vectors of length N
then you are looking for a vector z such that F(z) = 0 (I mean the vector of all zero components). This is a job for fsolve.
Alan Weiss
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