Chapter 11.5, Problem 1E

Interpretation Introduction

Interpretation:

To write the program to compute the correlation dimension of the Lorenz attractor. Reproduce the results in Figure $\text{11}\text{.5}\text{.3}$; then try other values of r, and check how dimension depends on r.

Concept Introduction:

An attractor that come out in a simplified system of equations which describes two-dimensional fluid flow of uniform depth is known as Lorenz attractor.

It is a system of ordinary differential equation.

Answer to Problem 1E

Solution:

The Matlab program is written to compute the correlation dimension of the Lorenz attractor.

The slope of the second figure depends on the value of correlation; hence, the dimension depends on r.

Explanation of Solution

Using Matlab, we can write a program to compute the correlation dimension of the Lorenz attractor.

$\begin{array}{l}\text{function loren3}\hfill \\ \text{clear;clf}\hfill \\ \text{global A B R}\hfill \\ \text{A=10;}\hfill \\ \text{B=8/3;}\hfill \\ \text{R=28;}\hfill \\ \text{u0=100*}\left(\text{rand}\left(\text{3,1}\right)\text{-0}\text{.5}\right)\text{;}\hfill \\ \left[\text{t,u}\right]\text{=ode45}\left(\text{@lor2,}\left[\text{0,100}\right]\text{,u0}\right)\text{;}\hfill \\ \text{N=find}\left(\text{t>10}\right)\text{;}\hfill \\ \text{v=u}\left(\text{N,:}\right)\text{;}\hfill \\ \text{x=v}\left(\text{:,1}\right)\text{;}\hfill \\ \text{y=v}\left(\text{:,2}\right)\text{;}\hfill \\ \text{z=v}\left(\text{:,3}\right)\text{;}\hfill \\ \text{plot3}\left(\text{x,y,z}\right)\text{;}\hfill \\ \text{view}\left(\text{158,14}\right)\hfill \\ \text{function uprime=lor2}\left(\text{t,u}\right)\hfill \\ \text{global A B R}\hfill \\ \text{uprime=zeros}\left(\text{3,1}\right)\text{;}\hfill \\ \text{uprime}\left(\text{1}\right)\text{=-A*u}\left(\text{1}\right)\text{+A*u}\left(\text{2}\right)\text{;}\hfill \\ \text{uprime}\left(\text{2}\right)\text{=R*u}\left(\text{1}\right)\text{-u}\left(\text{2}\right)\text{-u}\left(\text{1}\right)\text{*u}\left(\text{3}\right)\text{;}\hfill \\ \text{uprime}\left(\text{3}\right)\text{=-B*u}\left(\text{3}\right)\text{+u}\left(\text{1}\right)\text{*u}\left(\text{2}\right)\text{;}\hfill \end{array}$

By running the above Matlab code, we get the Lorenz attractor correlation as below:

This is the correlation dimension of Lorenz attractor.

Now consider the slope to be ${\text{d}}_{\text{corr}}\text{=2}\text{.05}$ and intersection $\text{c = -23}$.

Now use the below Matlab code to reproduce the figure $\text{11}\text{.5}\text{.3}$.

$\begin{array}{l}\text{d=2}\text{.05;c=-23;x=0:15;}\hfill \\ \text{plot}\left(\text{x,d*x+c}\right)\text{;}\hfill \end{array}$

Hence, this is the required curve.

The slope of the above figure depends on the value of correlation; hence, the dimension depends on r.

Conclusion

The correlation dimension of the Lorenz attractor is computed. The slope of the figure depends on the value of correlation; hence, dimension depends on r.