# phase portrait of system of differential equations

Is there a full classification of the phase portraits of the following systems of differential equations

$$\cases{ \dot x=a_{11}x+a_{12}y+a_{13}z \\ \dot y=a_{2 1}x+a_{22}y+a_{23}z\\ \dot z=a_{31}x+a_{32}y+a_{33}z}$$

and

$$\cases{ \dot x=f(x,y)\\ \dot y=g(x,y) }$$ Here $f(x,y), g(x,y)$ are polynomials of degree 3.

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The second system is a particular case of an open problem - A full classification of the behaviors of planar polynomial vector fields is not known at this time. Indeed, I believe that only the cases of $\deg(f),\deg(g) =1$ are known. In particular, very little is known about the existence, let alone the location of, limit cycles for polynomial vector fields. The investigation of such limit cycles is part of Hilbert's sixteenth problem, which is still unsolved. Not even the question of how many limit cycles are possible on a planar polynomial vector field with polynomials of degree $n = 2$ has been resolved. However, there is an example of a quadratic vector field with four limit cycles, seeShi Song Ling. As you might expect, then the case of $n = 3$ is even harder. For the longest time, the best possible example was 11 limit cycles, but as indicated in this paper (the top google hit), a new example with 12 limit cycles appears to have been discovered.