Lorenz (stability with Liapunov function)

[gop]

Homework Statement

Using the Liapunov function [tex]V=1/2(x^2+\sigma y^2 + \sigma z^2)[/tex], obtain conditions on sigma, rho, beta sufficient for global asymptotic stability of the origin in the Lorenz equation.

Homework Equations

The Lorenz equation

[tex]\dot{x}=\sigma (y-x); \dot{y}=\rho x-y-xz; \dot{z}=-\beta z + xy[/tex]

The Attempt at a Solution

[tex]\dot{V}=-\sigma (x^2+y^2+\beta z^2-(\rho+1)xy)[/tex]

Now i have to find conditions on beta and rho such the term in the brackets is positive , at least locally around (0,0,0). But I don't really know how to do that.

 

[mighty2000]

Thank you for your interesting question. I would like to provide you with some insights and guidance on how to approach this problem.

Firstly, it is important to understand the concept of a Liapunov function and its role in determining the stability of a system. A Liapunov function is a scalar function that is used to analyze the stability of a dynamical system. It measures the energy of the system and can be used to show if the system is stable, unstable, or chaotic.

In this case, the Liapunov function V is defined as V=1/2(x^2+\sigma y^2 + \sigma z^2). To determine the stability of the origin (0,0,0) in the Lorenz equation, we need to find conditions on sigma, rho, and beta such that V is positive and decreasing as the system evolves.

To do this, we can take the derivative of V with respect to time:

\dot{V}=x\dot{x}+\sigma y\dot{y}+\sigma z\dot{z}

Substituting the Lorenz equation into this expression, we get:

\dot{V}=-\sigma (x^2+y^2+\beta z^2-(\rho+1)xy)

Now, since we want \dot{V} to be negative for stability, we need the expression in the brackets to be positive. This can be achieved by setting the following conditions:

1. \beta > 0
2. \rho > 0
3. \sigma > 0

These conditions ensure that the energy of the system is decreasing and the origin is globally asymptotically stable.

In conclusion, to guarantee global asymptotic stability of the origin in the Lorenz equation, we need to have \beta > 0, \rho > 0, and \sigma > 0. I hope this helps in your understanding of the problem. Let me know if you have any further questions.