The Trapping Region of the Lorenz equations

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• Oliver321
In summary, the conversation discusses different methods for finding the trapping region of a nonlinear system of differential equations, specifically the Lorenz equations. One method involves using a Lyapunov function while the other involves converting to polar coordinates. The experts in the conversation note that using different Lyapunov functions can result in different trapping regions and there is no guarantee that a particular region is minimal.
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!

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
Office_Shredder said:
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?

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

Oliver321
pasmith said:
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?

Oliver321 said:
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
pasmith said:
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.

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?

1. What are the Lorenz equations and what is their significance?

The Lorenz equations are a set of three nonlinear differential equations that describe a simplified model of atmospheric convection. They were developed by meteorologist Edward Lorenz in the 1960s and have been widely studied in chaos theory. The equations have had a significant impact on the field of mathematics and have been used to model various complex systems.

2. What is the trapping region in the Lorenz equations?

The trapping region in the Lorenz equations is a subset of the phase space where the system's trajectory is confined. It is bounded by a set of critical points, also known as fixed points or equilibria, which are points where the system's behavior does not change over time. The trapping region is an important concept in understanding the dynamics of the Lorenz system.

3. How is the trapping region related to chaos in the Lorenz equations?

The trapping region is intimately connected to chaos in the Lorenz equations. In the original paper by Lorenz, he showed that the system exhibits chaotic behavior within the trapping region. This means that even small changes in the initial conditions can lead to drastically different outcomes, making the system unpredictable and sensitive to slight perturbations.

4. Can the trapping region be visualized?

Yes, the trapping region can be visualized using a phase space plot. In this plot, the three variables of the Lorenz equations (x, y, and z) are plotted against each other. The trapping region is represented as a bounded region in the plot, surrounded by the system's trajectory. This visualization helps to understand the behavior of the system and its sensitivity to initial conditions.

5. How does the size of the trapping region affect the behavior of the Lorenz system?

The size of the trapping region has a significant impact on the behavior of the Lorenz system. A larger trapping region means that the system has more space to explore, leading to a greater variety of possible trajectories and potentially more chaotic behavior. Conversely, a smaller trapping region restricts the system's dynamics, resulting in a more predictable behavior. The size of the trapping region can be controlled by adjusting the parameters in the Lorenz equations.

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