The Trapping Region of the Lorenz equations

[Oliver321]

I was dealing with nonlinear systems of differential equations like the Lorenz equations (https://en.wikipedia.org/wiki/Lorenz_system). Now there is a trapping region of this system defined by the ellipsoid ρx^2+σy^2+σ(z-2ρ)^2<R.
I wondered how this region is found and I found out that a Lyapunov function is used.
However, I tried to do it another way. I defined r=√(x^2+y^2+z^2). Now I thought: when d/dt r <0 the radial velocity points inward and so there is a trapping region. I calculated this but the result was completely different.
Why doesn’t it work with my way?

Thanks for every awnser!

 

[Office_Shredder]

In the trapping region, isn't dr/dt often bigger than zero? The particles move around and make loops around the attractor, but dr/dt isn't constantly shrinking while doing this.

 

[Oliver321]

In the trapping region, isn't dr/dt often bigger than zero? The particles move around and make loops around the attractor, but dr/dt isn't constantly shrinking while doing this.

Yes that’s right, but on the edge of the trapping region all arrows should point inward (or at least be tangential to the boundary), so dr/dt should be less or equal to zero?

 

[pasmith]

If you use different Liapunov functions, you will get different trapping regions.

 

[Oliver321]

If you use different Liapunov functions, you will get different trapping regions.

But why? With both methods I can get a minimal region, which is a trapping region. Because it is minimal, it has to be the same?

 

[pasmith]

But why? With both methods I can get a minimal region, which is a trapping region. Because it is minimal, it has to be the same?

How do you know the region is minimal?

All you can conlcude from looking at a particular Liapunov function is that the attractor is in some region, because once trajextories enter it they cannot leave. But there's always the possibility that a different Liapunov function would give you a stirctly smaller region, and hence a closer bound on the attractor.

 

[Oliver321]

How do you know the region is minimal?

All you can conlcude from looking at a particular Liapunov function is that the attractor is in some region, because once trajextories enter it they cannot leave. But there's always the possibility that a different Liapunov function would give you a stirctly smaller region, and hence a closer bound on the attractor.

Thanks for the answer!
So I thought that there is only one Liapunov function for a system. Like if i have a conservative system I can use a energy function to do the same things (and the Energy function is unique?). A liapunov function is a generalised energy function (at least so I was told). But if there are a few, this would be plausible.
Nevertheless the method with converting to polar coordinates should give a minimal region?