If you are working on let's say chemical reaction kinetics, x represents concentrations, which are real. If, for example, f(x) has three zeros, one real and the other two complex, then only the real zero makes sense as a set of concentrations representing an equilibrium.

For example, if you have x' = -(x3+x), you have the "equilibria" x = 0, x = ±i. A real-valued solution will tend to the x=0 solution. The other two equilibria won't play any role.

You would have to "complexity" your equation, putting x = u + iv. Then the real and imaginary parts satisfy, respectively, the real system

u' = -(u3 - 3uv2 + u)

v' = -(3u2v - v3 + v)

Here you have the same equilibria as before: u = 0 with v = ±1. But does v add any information or insight?

The problem here, if you are doing chemical reactions, is that there is a problem interpreting the imaginary part v. In all likelihood, only v=0 makes sense, in which case we're back to

u' = -(u3+u)

v' = 0

where the second equation has v(0) = 0, implying that v(t) = 0 for all t, and where the first equation is your original equation before trying "comlexifying" your equation.

So this extension of the problem didn't gain anything new, and my guess is that "complex equilibria" don't matter, as they have no interpretation in terns of chemical concentrations.   

Hello,

sometimes it is possible to rewrite a real 2-dimensional equation as a scalar complex equation. Using the complex variant might have some advantages, like simpler expressions for (discrete or continuous) rotational symmetries.

Another idea (as far as I remember born by H.B. Keller) is, that branches of complex equilibria might connect branches of real equilibria. So it helps you to detect other solution branches, which might be hard to detect if you restrict yourself to real solutions.

Maybe you find more advantages in the papers, which investigate these equilibria.

I do not understand  clearly if you refer to equilibrium points of linearized systems  with nonzero imaginary component. if so , there are typically a) the center with pure imaginary components  - local trayectories around it are circles or ellipses , b) the focus  with nonzero real part and imaginary components - the local trajectory being a logarithmic spiral.  In the fiirst case , the solution is an oscillation , in the second one a dampled oscilaltion. This description is also useful for nonlinear systems - described by their linearized versions- for local deviations from the equilibria. A restriction is that the real part of the eigenvalues be nonzero for the validity of the linearization ( otherwise, the second order terms  would be the dominant ones in the dynamics ) . This excludes centers  (and also nodes with one ofthe eigenvalues of the equilibrium of the linearized system  being zero).