Dynamics of a three dimensional system

Question

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?

Answer 1

A few hints that should let you complete the picture:

• $r'=(1-r^2)r$, where $r=(x^2+y^2+z^2)^{1/2}$;

• $\theta'=1$ for the angle in the $xy$-plane;

• the $z$-axis is invariant, satisfying the equation $z'=(1-z^2)z$.

In particular, the sphere of radius $1$ centered at the origin is invariant.

Answer 2

From $r \dot r =x\dot x + y\dot y +z\dot z$

$r \dot r =x\dot x + y\dot y +z\dot z$

$r\dot r = -x^4-2x^2y^2-2x^2z^2-y^4-2y^2z^2-z^4+x^2+y^2+z^2$

$ r\dot r = (x^2+y^2+z^2) (1 - (x^2+y^2+z^2)) = r^2(1-r^2) $

$\dot r = r(1-r^2) $

In terms of stability, r=0 is a repeller and r=1 is an attractor.

Looking at $\dot \theta = (x\dot y-y \dot x)/r^2$

$ \dot \theta = (x^2+y^2)/r^2 = 1 $

Additionally, the ODEs are invariant under the transformation (x,y,z,t) ==> (-x,-y,z,t)

which shows that solutions are symmetric about the z axis.

Thus, based on this analysis, trajectories for the system will go out to the sphere of radius 1 and rotate counterclockwise with respect to the xy plane. There is no periodicity as there is no time reversal symmetry about the z axis.