I have a dynamical system in three dimensions given by:
$\dot x = (1-x^2-y^2-z^2)x+xz-y$
$\dot y = (1-x^2-y^2-z^2)y+yz+x$
$\dot z = (1-x^2-y^2-z^2)z-x^2-y^2$
I analyzed the system by first finding the fixed points which are at (0,0,0) , (0,0,1) and (0,0,-1).
Next, I found the Jacobian of the system and for each fixed point, I computed the eigenvalues of the Jacobian matrix.
For the fixed point (0,0,0), $\lambda = 1, 1+i, 1-i$
For the fixed point (0,0,1), $\lambda = -2, 1+i, 1-i$
For the fixed point (0,0,-1), $\lambda = -2, -1+i, -1-i$
Hence, (0,0,0) is unstable since the eigenvalues are positive. (0,0,1) is unstable since the real part of the complex eigenvalue is positive. (0,0,-1) is negative since the eigenvalues are negative.(i.e. real part of complex eigenvalues are negative).
Is there anything else I can say about the long term behavior of this system?