If we have a nonlinear first order ODE system,
$$x'(t) =f (x, y)$$
$$y'(t) =g (x, y)$$
and we approximate it to a linear system
$$x'(t) = ax + by$$
$$y'(t) = cx + dy$$
and we get for a critical point $\vec{x_0}$ pure imaginary eigenvalues, $\lambda=\pm qi$, it is said that this critical point will be a center of the linear system, but that it may be either a center or a focus of the nonlinear system.
How can we know if it is a center or a focus of the nonlinear system?
For instance, if we have the system
$$x'(t) = x + y$$
$$y'(t) = 2x -2x^2-y$$
There is a center of the linear approximation at the critical point $(3/2,-3/2)$. How could we determine if it is a center or a focus of the nonlinear system?
It is easy to get the first integral, but not to draw it (without computer).