I am very new to the nonlinear dynamical-systems. I wonder if there is a method to determine when a limit cycle situation will occur in the non-linear dynamical system, which is described by the differential equations: $$ \frac{d}{dt}\mathbf{x}=f(\mathbf{x}), $$ where $x=(x_1,x_2,\cdots,x_8)^T$, the nonlinear term is just $Vx_{2}x_j$ for $dx_j/dt=f_j(x) (j=5,6,7,8)$.
As far as I know, before analyzing the stability of the system, we need to know the fixed point $x^*$, which satisfied $f(x^*)=0$. But here the solution $x^*$ is hard to obtain, or I can only obtain numerical solution through numerical method, such as RK4. In the numerical calculation, there will be limit cycle situation (sometimes just stable, determineing by my parameters), which means I can not get the fixed point $x^*$ from the Rk4.
$\mathbf{Question}$: I want to know if there is method to judge when the limit cycle occure, in the case where I cannot know the fixed point $x^*$ under certain parameters.
Thanks!