cancel
Showing results for 
Search instead for 
Did you mean: 
cancel
Showing results for 
Search instead for 
Did you mean: 

3D Plot of Lorenz Attractor

RogerMansfield
1-Newbie

3D Plot of Lorenz Attractor

The "strange attractor" of E. N. Lorenz is documented in many places in the literature of fractals and chaos. See, for example, references [1] through [4].

But the Mathcad 13 example attached below is based upon an image that I saw in Mathsoft's sales brochure for Mathcad PLUS 6.0 for Macintosh back in late 1996. So far, I have not found the Lorenz attractor, as a worksheet, in any of the Mathcad Resource Centers that were bundled with Mathcad in the mid-to-late 1990s (but perhaps I did not look hard enough). The Mathcad worksheet itself must exist somewhere in the archives of Mathsoft (now PTC), since the worksheet did in fact appear in the printed sales brochure from 1996 (I went back to my collection of sales literature and verified this).

My own example worksheet has a number of control parameters that allow you to experiment with the numerical integration process. Further, the attractor is illustrated in the worksheet via a 3D plot. You can click on the 3D plot (in the worksheet, not in the animation of the worksheet), and Mathcad will let you rotate the image about any desired axis, so as to examine the attractor from any aspect.

The Lorenz attractor is an example of chaotic dynamics in 3-dimensional space. It is the solution to an initial value problem consisting of (a) a system of three non-linear, first-order ODEs and (b) a vector of initial conditions.

The worksheet sets up the system of three first-order ODEs, and then solves them using Mathcad's adaptive Runge-Kutta numerical integration function, Rkadapt.

The process of simultaneously solving systems of first-order, ordinary differential equations (which may be linear or nonlinear) is at the core of the engineering discipline known as state space analysis. I myself have used the state space approach to solve problems in the determination of artificial Earth satellite orbits, comet and asteroid orbits, and the trajectories of interplanetary spacecraft. Some of my work here can be found by clicking on the "Mathcad Worksheets by Astroger" link at http://astroger.com/.

If you do not have Mathcad installed on the machine that you are presently using to view this document, then try clicking on the PDF version of the worksheet, also attached below.

References

[1] Drazin, P. G., Nonlinear Systems, Cambridge University Press (1992), Chapter 8.

[2] Korsch, H. J. and Jodl, H.-J., Chaos: A Program Collection for the PC, Springer-Verlag (1994), pp. 270-274.

[3] Robbins, Judd, Fun with Fractals, Sybex, Inc. (1993), Chapter 14.

[4] Stevens, Roger T., Fractal Programming in Turbo Pascal, M&T Publishing, Inc. (1991), Chapter 4.

0 REPLIES 0
Announcements
Check out the latest
Mathcad Tip
"PTC Mathcad 15 / Prime 1-6 Update."