Knowing dynamical equations, one can straightforwardly solve them algebraically to find the fixed points and then anticipate the field behavior near them by finding the eigenvalues. But there are some trajectories that appear as an attractor paths i.e. draw inwards neighbor field flows and force them to follow a same path. Is there any systematic method to find such attractor tracks? For example, Let's the equations be $$ \frac{dx}{dt}=x^2 + x y^2 \; and\; \frac{dy}{dt}=y + 2 x^2 - y^2 $$ Then the fixed points are $(0,0)$, $(0,1)$ & $(-1,-1)$ where you can see their special behavior in the below phase portrait diagram .
But one also recognizes the other behavior in which the flows try to reach two particular trajectories in the upper half plain. What is the equation of such attractor trajectories in general? That is the question.
If I understand your question correctly, from looking at the trajectories of that system, you have got the impression that some trajectories are kind of converging to a specific trajectory, and you want to know what that trajectory is.
The problem is, you can't guess the general behavior of a dynamical system just by looking at its stream-plot in a finite interval. But before discussing this matter, let's take a look at the system's equation: $$\begin{align}\dot{x}&=x^2+x y^2\\ \dot{y}&=y-y^2+2x^2\end{align}$$ which implies $$\frac{dy}{dx}=\frac{y-y^2+2x^2}{x^2+x y^2}$$ Now if you do a qualitative analysis on this equation, you can see that the terms with highest degree on the numerator and denominator are $2x^2$ and $xy^2$, respectively. Which implies that $\frac{dy}{dx}$ at large values of $x,y$ and $x\neq 0$ will be zero.
So the problem was, you did a quick judgement and generalized a local trend to a global one. These diagrams are self-evident: