Understanding Lorenz Equations & Stability: A Homework Guide

In summary, the conversation is discussing a problem related to the Lorenz Equations and finding fixed points and stability using different methods. The question is also asking about how to interpret a derivative and what a period-2 orbit means. The person is asking for clarification and an example to better understand the problem.
  • #1
piareround
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0

Homework Statement


http://imageshack.us/photo/my-images/535/newpicture3.jpg/


Homework Equations



Honestly i used the equation in Strogatz 10.3.3 so I am not sure how to do it for others.


The Attempt at a Solution


http://imageshack.us/photo/my-images/406/worke.png/
http://imageshack.us/photo/my-images/406/worke.png/
As you can see I found the fixed points by treating the problem as a map instead of as somehow related to the Lorenz Equations. I am really not sure how they want us to use the Lorentz euqations to do the same thing.
Moving on though I found the stability like I normally do by taking a derivative and setting that equal to zero. However, in this case I am really not sure how to interperate that derivative. Does anyone know after find the derivative you supposed to show stability for Lorenz systems?
Finally I am not really sure what they mean by period doubling orbit. I know what period doubling is and I know an orbit is like a circular trajectory, but I am really confused at what the two together are supposed to mean. Could anyone help clarify what the last portion of the question is asking and how you are supposed to show it? Prehaps by showing a similar example?
 
Last edited:
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  • #2
Sorry if that last post is confusing, I guess the question I really need answer is: what is period-2 orbit?
 

1. What are the Lorenz equations?

The Lorenz equations are a set of three nonlinear differential equations that describe the dynamics of a simplified mathematical model of atmospheric convection. They were developed by meteorologist Edward Lorenz in the 1960s and are commonly used in chaos theory and nonlinear dynamics.

2. What is the significance of the Lorenz equations?

The Lorenz equations are significant because they are one of the earliest and most well-known examples of a chaotic system. They demonstrate how small changes in initial conditions can lead to drastically different outcomes, making long-term prediction of weather patterns and other complex systems difficult.

3. How do the Lorenz equations relate to stability?

The Lorenz equations can exhibit both stable and unstable behavior depending on certain parameter values. In particular, there is a critical value of one of the parameters known as the Rayleigh number, at which the system undergoes a bifurcation from stable fixed points to chaotic behavior. Understanding this transition is important for studying the stability of other systems.

4. Can the Lorenz equations be solved analytically?

No, the Lorenz equations do not have a closed-form analytical solution. They can only be solved numerically using methods such as Euler's method or the Runge-Kutta method.

5. How are the Lorenz equations used in real-world applications?

The Lorenz equations have been used to model a variety of physical and biological systems, such as ocean currents, chemical reactions, and population dynamics. They have also been applied in fields such as economics, sociology, and neuroscience to study complex behavior and patterns in these systems.

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