Planar Motion of Dyanamical System

Question

Given the system

$ \dot x = y^2 - z^2 + xy $

$ \dot y = 2x^2 - y^2 - xz $

$ \dot z = 2z^2 - 4x^2 - 2xy + 2xz $

how do I show that the motion of this system is along a plane and what can I say about its long term behavior?

First, the obvious fixed point where $ \dot x = \dot y = \dot z = 0$ is at the point $(0,0,0)$. The eigenvalues of the Jacobian matrix evaluated at this point are just $ λ=0$, so the fixed point is stable. This is fine, but I'm unsure how to prove that this system's motion lies on a plane.

Answer 1

Let $w = z + 2y + 2x$. You will find that $\dot{w} = 0$ using your system. This implies that $w$ is constant for any solution. Notice that $w$ is a linear function of the coordinates $(x,y,z)$, and this means that the motion of the dynamical system is always orthogonal to the vector $(2,2,1)$ and hence reside in planes.