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.

Please, provide any reference.

-

1 Answer

The first system 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.

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.

-
@Christopher Thank you. 1. I see and know how to do it. But I had asked about any reference for this case. 2. Unfortunately all links dont work. – Melania Oct 28 '12 at 15:23
@Melania, sorry, hopefully the links work now. I'm not sure whether you can find a reference specifically for the cubic case, since the theory for all linear systems is more-or-less the same. – Christopher A. Wong Oct 28 '12 at 20:27
@Christopher, no problem, give me a reference to classification of the phase portraits for arbitrary linear system – Melania Oct 28 '12 at 21:15
This smells to me close to HW from a systems course. What is your motivation Melania? Close this question? – Richard Montgomery Oct 29 '12 at 1:18
@ Richard. No of cource. I just want to know what is the sitiation with the complete classification of the the phase portraits for the 3x3 linear system.Is it done or not? If it already done, then please provide reference. I hope that it is done. – Melania Oct 29 '12 at 8:14