Question

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

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

and

2. $$ \dot x=f(x,y), \dot y=g(x,y)? $$ Here $f(x,y), g(x,y)$ are polynomials of the degree 3.

Please, provide any reference.

Answer 1

1. This is a linear system, so its phase portrait can be fully determined from the eigenvector structure of the associated matrix. In particular, phase portraits for such systems can be classified according to types of eigenvalues which appear (the sign of the real part, zero or nonzero imaginary parts) and the dimensions of the generalized eigenspaces.

2. This is 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, an example exists of a quadratic vector field with four limit cycles. 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.