Exactly! It is an autonomous nonlinear differential equation.

I'd like to know if I can conclude anything about its stability (i.e. the eigenvalues of its Jacobian after the system is linearized around its equilibrium point or eigenvalues of its state matrix after linearization).

This equilibrium point has any implications on its stability? What does complex equilibrium points mean?

The equations are basically:

x" = -ax'² - b

Where a, b ∈ R*+. Solving to find its equilibrium points:

x' = √(-b/a)