Here we'll talk about both linear and nonlinear planar systems of the form
$$ \begin{aligned} \dot{x} & = f(x,y) \\ \dot{y} & = g(x,y). \end{aligned} $$
A separatrix is the boundary that separating two modes of behavior in a dynamical system.
Trajectories that start and end at the same fixed point are called homoclinic orbits.
Trajectories that connect two fixed point are called heteroclinic orbits.
A hyperbolic fixed point is a fixed point for which the real part of both eigenvalues is non-zero.