Determine the omega limit set for every point in $\Re^3$ for this dynamical system

Question

The dynamical system is as follows

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

Now by looking at the two dimensional system with only $x$ and $y$ I know that every point here is just a spiral. (centre manifold) http://m.wolframalpha.com/input/?i=streamplot%5B%7B-y%2C+x%7D%5D

So when adding in the $z$ part to make it 3 dimensional I will get like a spiral orthogonal to every point on the spiral in the two dimensional plane. (I hope I am making sense here).

I am struggling to fully visualise how it would look but what I think is that the omega limit points will be the circle around each point in the two dimensional plane.

But I don't know how to explicitly say what the omega limit set for each point is.

Edit: I will attempt to make myself more clear here.

But the stationary points of the DS can be found at $(x^*, x^*, z^*) = (0,0,z)$ which is basically the $z$ axis.

Also I know that for each point in the $x,y$ plane that their is a helix in the $z$ direction.

So now what I am thinking is that as time progresses each point will be moving towards the stationary point. Hence the z axis is the omega limit set?

Answer 1

I don't know what the limit set is but you can simplify this ODE by the following steps.

Note that the first two equations are uncoupled from the last one. Hence, divide both equations to obtain

$$dx/dy = -y/x \implies xdx = -ydy \implies x^2+y^2=c.$$

For certain intitial conditions $(x_0,y_0,z_0)$ we have $c = x^2_0+y^2_0$.

Now, use $x^2+y^2=c$ to substitute $-y^2=x^2-c$ to obtain the last ODE as

$$dz/dt=-x^2+x^2-c=-c \implies z(t)=-ct+c_2=-(x_0^2+y_0^2)t+c_2.$$

Again we can use the initial conditions to determine $c_2=z_0$. Hence,

$$z(t)=-(x_0^2+y_0^2)t+z_0$$. Can you complete it from here?

Answer 2

You can be more specific (and use better terminology) than "spiral". The trajectories of the two dimensional system are circles centered on the origin, except that the origin is a stationary point. And the trajectories of the three dimensional system are helixes lying over those circles, travelling at a constant downward speed on each such helix, except that every point on the $z$-axis itself is a stationary point.

So, now ask yourself: As you travel at constant downward speed along a helix, what's the omega limit set, i.e. what points do you accumulate on? And as you sit unmoving on a stationary point, what's the omega limit set, i.e. what points do you accumulate on?

Added: Here's one way to pose the question to yourself regarding the omega limit set of a trajectory.

Imagine that you are an ant walking on one of these helix trajectories. You start walking downward, descending lower and lower and lower, as your $z$-coordinate decreases at a constant rate. Down, down, down, lower, lower, lower down the helix. You walk past some point $P$ (wave at it!), and continue on downward, lower, lower, lower, never returning anywhere near $P$ again (goodbye $P$!). Are you accumulating on $P$? Is $P$ in the omega limit set? If you are accumulating on $P$ then yes, $P$ is in the omega limit set. If you are not accumulating on $P$ then no, $P$ is not in the omega limit set.