Understanding Lorenz Equations & Stability: A Homework Guide

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SUMMARY

The discussion focuses on the application of Lorenz equations in understanding stability and fixed points in dynamical systems, specifically referencing Strogatz's equation 10.3.3. The user expresses confusion regarding the interpretation of derivatives in the context of Lorenz systems and seeks clarification on the concept of period doubling orbits. The key takeaway is that to analyze stability in Lorenz systems, one must compute the derivative of the system and understand its implications for fixed points and periodic orbits.

PREREQUISITES
  • Understanding of Lorenz equations and their applications in dynamical systems.
  • Familiarity with Strogatz's "Nonlinear Dynamics and Chaos" and its equations.
  • Knowledge of derivatives and their role in stability analysis.
  • Concept of periodic orbits and period doubling in dynamical systems.
NEXT STEPS
  • Study the derivation and implications of fixed points in Lorenz systems.
  • Learn how to apply stability analysis using derivatives in nonlinear dynamics.
  • Research the concept of period doubling orbits in chaotic systems.
  • Explore examples of Lorenz attractors and their stability characteristics.
USEFUL FOR

Students and researchers in mathematics, physics, and engineering who are studying dynamical systems, particularly those interested in chaos theory and stability analysis of Lorenz equations.

piareround
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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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Sorry if that last post is confusing, I guess the question I really need answer is: what is period-2 orbit?
 

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