Do we have to consider complex states when trying to find equilibrium points analytically?

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If you want to find the equilibrium points analytically for problem 1b of HW2, you can write a 5th order polynomial in terms of z1 or in terms of z2. You will get 2 complex roots and 3 real. Since you are solving for equilibrium points, which are states where dz/dt = 0 and all states have to be real to make physical sense (could represent position, velocity, population density, etc.), you only use the real roots of the 5th order polynomial. In general, we always ignore the complex roots. Phaseplot, pplane7 and other numerical packages will only plot real states.

Of course, if eigenvalues is what you are trying to find, then you have to consider complex eigenvalues as well. Remember that wherever the eigenvalues of the system dynamics matrix A are located in the complex plane will determine stability and affect system performance. --Soto 12:32, 13 October 2008 (PDT)