This Compact Analog Computer Cleverly Calculates Lorenz Attractors for Convection Simulation
With no digital processor in sight, this single-purpose computer outputs ethereal Lorenz attractor solutions to convection models.
Pseudonymous student "CheeenNPP" has built a computer with a difference: it's analog, rather than digital, and serves to solve Lorenz system equations — offering butterfly-like Lorenz attractor solutions as its output.
"One of the main purposes of analog circuits is to solve mathematical problems, such as building a circuit corresponding to a non-linear differential equation and analyzing the phase plane characteristics of it by observing its output voltage with an oscilloscope or analog plotter," CheeenNPP explains of the project. "I will take a famous non-linear differential equation, the Lorenz Attractor, as an example, and show the whole process of solving it with an analog circuit."
A system of three equations created by Edward Lorenz, Ellen Fetter, and Margaret Hamilton in 1963, the Lorenz system offers a simplified mathematical model of atmospheric convection based on a two-dimensional fluid layer which is warned from below and cooled from above. It's most famous for the Lorenz attractor solution family, which produces a pattern not unlike the wings of a butterfly — and it's these solutions CheeenNPP's analog computer is built to provide.
"Integral: use [an] inverting integrator circuit, but temporarily discard the resistor, which is equivalent to integrating the input current and multiplying by -1," CheeenNPP explains of how their creation works. "Multiplying by -1: use the inverting amplifier and set the two resistors with the same value. Addition: use current summation instead of voltage, so the form is simpler. Multiplying by a constant k: use a resistor with a resistance of 1/k, the voltage across it is converted into a current and output as a result. Multiplication: implemented using a dedicated voltage multiplier."
Following LTSpice simulation to prove the concept, CheeenNPP set about building it on a prototyping board — resulting in a surprisingly compact creation with a maze of solder and jumper-wire links beneath. Connected to an oscilloscope, the familiar winged pattern of a Lorenz attractor is immediately visible.
"The phosphor afterglow of the analog oscilloscope […] has a very illusory and ethereal beauty," CheeenNPP says, "just like the elusive chaos implied by the Lorentzian attractor."
A circuit diagram and more information on the project is available on CheeenNPP's Hackaday.io page.