Design Files for Download:

Schematic Diagram in PDF format

PCB Gerber Bottom Layer

PCB Gerber Top Overlay

PCB Gerber Bottom Solder Mask

PCB NC Drill File

LTspice ASC File

LTspice PLT File

The Lorenz system, originally discovered by American mathematician and meteorologist, Edward Norton Lorenz, is a system that exhibits continuous-time chaos and is described by three
coupled, ordinary differential equations. The Lorenz system is related to the Rössler attractor, but is more
complex, having two quadratic nonlinearities rather than just one, and generates a chaotic attractor having two lobes rather than just one. The equations are:
*
**x*, *y* and *z* state solutions within the linear operating range of the operational amplifiers
and the two analog multiplier chips running on +/-15V supply rails, a scaling (down) factor of 5:1 was used. Using standard value resistors, the *s*, *r* and *b*
coefficient values were set as close as practical to the values originally studied by Edward. Coefficient *b* scales to 1Meg / (R4 + R5). Coefficient *r* scales to 1Meg / (R8 + R9)
and coefficient *s* scales to 1Meg / R, where R = R12 = R13. The global scaling factor of 5:1 is determined by 0.1Meg / R, where R = R3 = R10. The global scaling factor is
referenced to 0.1Meg rather than 1Meg due to the transfer function of the analogue multiplier integrated circuits, which divide the product of their X and Y input signals by 10.

dx/dt = -sx+sy

dy/dt = rx-y-xz

dz/dt = -bz+xy

The behaviour of the system is chaotic for certain value ranges of the three coefficients, *s*, *r* and *b*. The values originally studied by Edward Lorenz were
*s*=10, *r*=28 and *b*=2.67 (8/3). An excellent and much more thorough introduction to the Lorenz system, with references, is available at the Wikipedia page
here.

While the Lorenz attractor is readily simulated with iterative, discrete-type digital computation techniques on a modern desktop P.C., using software packages such as MATLAB, or even in SPICE with a little more difficulty, it is also readily simulated with simple electronics hardware conforming to a much older concept; that of continuous analog computation. A complete electrical analog of the Lorenz attractor, as described by the three differential equations above, can be implemented with an interconnection of just three distinct, basic and common circuit building blocks; namely the summing amplifier, the integrator and the analog multiplier. Here is my version of a circuit that does just that:

In order to keep the computed and continuously-time variableThe phase space of the Lorenz attractor is mapped in the minimal three dimensions required for continuous-time chaos by the three computed states, *x*, *y* and *z*.
The famous "owl face" diagram of the Lorenz attractor is produced by neglecting the *y* state and plotting the *z* and *x* states in two dimensions. This is shown in the
oscilloscope picture incorporated into the schematic diagram above. The *x* state is plotted on the horizontal axis and the *z* state on the vertical. However better and
more instructive views can be had by using transformation techniques working with all three states to generate three-dimensional projections on a two-dimensional display.
The series of oscilloscope CRT photos pictured immediately below show three-dimensional projections of the Lorenz attractor at various angles of 2-axis rotation. These displays were
generated by my three-dimensional projective unit, with the Lorenz attractor circuit described here being the x-y-z signal
source.

I produced a small PCB for the entire circuit as depicted schematically above; the Gerber files of which can be downloaded via the links at the top of this page, along with the LTspice simulation files. The operation of the circuit is simulated in SPICE with a simple transient analysis:

Well, that's it. In conclusion; a rather simple and unusual function generator that makes for a good, demonstrative introduction to analog computation and chaos theory - and a fun weekend project as well.