A REVIEW OF LORENZ’ WORK
We discussed earlier that Lorenz obtained a strange attractor that resembled a butterfly when he solved the following set of equations:
dx/dt = s(y – x)
dy/dt = rx – y – xz
dz/dt = xy – bz
The values of the coefficients that he used were s = 10, r = 28, and b = 8/3 so his equations ended up as
dx/dt = 10(y – x)
dy/dt = 28x – y – xz
dz/dt = xy – (8/3)z
Lorenz laboriously solved these nonlinear differential equations on an early digital computer which was very primitive by today’s standards. He then plotted the results using phase-space techniques and obtained the butterfly strange attractor.
The equations can be solved much more easily (and accurately enough for our purposes) by applying the analog computer techniques we discussed previously. This is the approach we will take.
AN ANALOG COMPUTER CIRCUIT FOR SOLVING THE LORENZ EQUATIONS.
(Click on the image to enlarge it.)
The analog computer circuit shown above that we will use was developed by Paul Horowitz at Harvard University. A somewhat larger copy of this schematic diagram, together with the paper Paul wrote describing its operation, can be viewed by clicking on the link below.
Before we discuss this circuit, it should be noted that Paul Horowitz and Winfield Hill have written a truly excellent book on electronics entitled “The Art of Electronics.” This book should be on the bookshelf of anyone who is seriously interested in learning electronics. I recommend it HIGHLY.
BACK TO THE LORENZ ATTRACTOR CIRCUIT
The Lorenz attractor circuit designed by Paul Horowitz is ideally suited for both computer simulation and for building in the laboratory. I have both built and simulated this circuit and will discuss both approaches in some detail.
First, we will see how Paul’s analog computer circuit for solving Lorenz’ equations can be simulated and the butterfly strange attractor can be displayed using LTspice.
In my simulation, I deviated from Paul Horowitz’ circuit by using AD633 analog multipliers, rather than MPY634. Also, I used LT1057 op-amps rather than LF412. I did this solely due to the availability of the LTspice models for these components.
Let me mention, again, that I obtained the LTspice model for the AD633 from the www.analog-innovations.com) website by Jim Thompson.
The circuit for my Lorenz equations simulation is shown below. I have included the Spice model for the AD633 on the schematic and have used LT1057 op-amps so that you can simulate my circuit using my .asc file below without the need to add any additional components to the LTspice library.
However, if you plan to design and simulate your own circuits using the AD633 analog multiplier, be sure and add Jim Thompson’s LTspice model.
Below is a link to Jim Thompson’s website. Go to his “Device Models & Subcircuits” page to download his model for the AD633. If you get the AD633 LTspice model from other sources, you may have convergence problems.
Below is the circuit (based on Paul Horowitz’ work) I simulated to generate the Lorenz butterfly attractor.
This schematic for my Lorenz attractor circuit was used to generate the following .asc file.
Remember to drop the .txt follower when you download and save the file so that you end up with a file named MY_LORENZ.asc
Here is the “Lorenz Butterfly” that I obtained with my simulation.
BUILDING THE LORENZ BUTTERFLY CIRCUIT
Now, we will discuss how to use three op-amps and and a pair of analog multiplier integrated circuits to actually build this analog computer circuit in your lab. Once you have built the circuit, you can display the Lorenz butterfly strange attractor on an oscilloscope.
We will use Paul Horowitz’ circuit, which we saw above, but we will make two component changes. The MPY634 analog multiplier is obsolete (and is very expensive even if you can find it on the surplus component market). Consequently, you will want to use the much less expensive AD633 analog multiplier in its place.
The MPY634 is a 14-pin integrated circuit while the AD633 is an 8-pin device. Consequently, I am providing you with the AD633 datasheet so that you can see how the pins are numbered.
When I built this Lorenz circuit, I used TL082 op-amps. The pin numbering system is the same on most op-amps. Nonetheless, I am again including the TL082 datasheet for your reference.