Given the following dynamical system:
$ \dot x = -6x^2+yz+x-1 $
$ \dot y = 4xz-3y^2+y-2 $
$ \dot z = 9xy-2z^2+z-3 $
What can you say about its long term behavior?
Attempt:
First, finding the fixed points.
There is only one real solution to $x'=0$, $y'=0$ and $z'=0$ and this is at the point $(1,2,3)$. At this point, the eigenvalues of the Jacobian matrix are $\lambda = 1,-17 $ Because of the positive eigenvalue, this fixed point is unstable.
I have also run into fixed points for the above system but they are complex. Do these complex fixed points have an influence on the dynamics of the system?
The change of variables $x \to \xi$, $y \to 2 \eta$, $z \to 3 \zeta$ yields the system \begin{align} \dot{\xi} &= f(\xi) + 6 \eta \zeta, \\ \dot{\eta} &= f(\eta) + 6 \xi \zeta, \tag{1}\\ \dot{\zeta} &= f(\zeta) + 6 \xi \eta, \end{align} with \begin{equation} f(x) = -6 x^2 +x-1. \end{equation} Not only is the vector field of $(1)$ conservative (its curl vanishes), it is also invariant under all permutations of the triple $(\xi,\eta,\zeta)$.
This type of systems has been studied extensively by Martin Golubitsky. A good source is
M. Golubitsky, I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002.
Of particular interest is section 3.4 in chapter 3, on rings of cells.
Addition: The symmetry suggests another coordinate change. Note that the linear combination $\xi+\eta+\zeta$ is invariant under permutations of $(\xi,\eta,\zeta)$. Therefore, introducing the (suitably normalised) coordinates \begin{align} X &= \frac{\xi - \eta}{\sqrt{2}},\\ Y &= \frac{\xi+\eta - 2 \zeta}{\sqrt{6}},\\ Z &= \frac{\xi+\eta+\zeta}{\sqrt{3}}, \end{align} yields \begin{align} \dot{X} &= X(1 - 6 \sqrt{3} Z),\\ \dot{Y} &= Y(1 - 6 \sqrt{3} Z),\\ \dot{Z} &= Z-\sqrt{3} - 3 \sqrt{3}(X^2+Y^2). \end{align} The equation for $Z$ suggests the introduction of polar coordinates $X = R \cos \theta$, $Y = R \sin \theta$, yielding \begin{align} \dot{R} &= R(1 - 6 \sqrt{3} Z),\\ \dot{\theta} &= 0\\ \dot{Z} &= Z-\sqrt{3} - 3 \sqrt{3}R^2. \end{align} So, the three-dimensional system is reduced to a planar system.