A Concise Guide to Chaotic Electronic Circuits

Chaos is a remarkable phenomenon occurring in many nonlinear systems, where the deterministic nature of the structure conjugates with the irregularity of the behavior. (Preface, first sentence)

The first sentence in this gem of a book is not only eloquent but brought back memories of my first introduction to chaos. In a Nova episode many years ago, as I recall, the narrator described how Edward Lorenz discovered chaos. The story was he ran a system of nonlinear equations and stopped the program for some reason. Later he restarted the program and decided to set it back to an earlier time. He typed in the values of the system variables for that past time and let the program run. At first, the output was what he saw earlier from his program. However, after a short time, the system outputs diverged and no longer matched what he had seen earlier. The narrator said Lorenz had not typed in all the digits when he restarted the program, just four or five digits, and hence the divergent output showed sensitive dependence on initial conditions.

More recently I heard a slightly different take on this story. He ran a nonlinear system of equations and he stopped and restarted the program only to see divergent system outputs. But, the new twist is he typed in all the digits he knew for the states. He did not leave out any digits. That was not the issue. The issue with the numbers was the internal computer system representation. The variables were represented internally in such a way that not all of the precision was displayed. When Lorenz typed in the numbers, he could not know the full precision of the number representation. It was this missing precision, so to speak, that caused the divergence and displayed the sensitive dependence. It’s a small twist to the story but small changes can have big effects when the discussion is chaotic systems.

The earliest consideration, I think, of chaotic behavior is Henri Poincaré’s study of celestial mechanics. The Lorenz story reflects just how long current mathematicians and scientists have been interested in chaos. Lorenz’s discovery dates to 1963. One of the most popular books on chaos, James Gleick’s Chaos: Making a New Science first appeared in 1987 and was revised in 2008. A search for “chaotic systems” in Google shows “about 985,000 results.” Chaos is big science and has lots of attention.

What does this lovely book offer that other books, Internet searches, and magazines do not? A lot, it turns out.

This book takes the science and moves it from a programmer’s world to your hands in the form of wires and circuits. The book is a compendium of electronic circuits one can easily build, study with an oscilloscope, play with resistors and alter operational amplifiers. It’s a hardware and experimental guide to chaotic behavior in electronics you build.

The authors begin with an introduction to the building blocks of circuits. These include operational amplifiers, adders, resistor-capacitor integrators, analog multipliers, time-delays, and (my favorite) negative resistors. Circuit diagrams, component values, and behavior are all richly described, easy to follow, and come with pictures, to boot. Yes, you need to know what a resistor is, and what a circuit board is, but that’s about it. The authors walk you through the other details with clear prose and diagrams.

The next part of the book is collection of chaotic systems you can build to see chaotic behaviors. The book covers the jerk circuit, Chua’s circuit (famous for the double scroll), Lorenz system (famous as an early example and even recently proposed for waveforms in radar [1]), Hindmarsh-Rose neuron, Duffing system, Van der Pol circuit, and a time-delay chaotic circuit. There are more in the book. You can build these circuits, connect an oscilloscope, and voilà, see chaotic behavior. The authors show pictures of the circuits and the oscilloscope traces so you know when you have it right.

The last two chapters are about field programmable analog arrays (FPAA) to implement Chua’s circuit and CO₂ laser dynamics, and synchronization of chaotic circuits. These chapters were a bit weak. I would have liked to see more on FPAAs; I don’t know much about them and some introduction to these devices would have made the discussion more complete.

Synchronization is a fascinating idea and brings to mind coupled oscillators. The fact you can synchronize chaotic systems is marvelous and more details would have been helpful.

Despite these weaknesses, the aim of the book is a guide to build circuits and experiment. On that score this book does not disappoint. I recommend it for any hobbyist, mathematician, or engineer who wants to experience chaos beyond a mere simulation on a computer screen that we usually see.

Reference:

[1] Willsey, et al., Selecting the Lorenz Parameters for Wideband Radar Waveform Generation, World Scientific, May 2010, http://hdl.handle.net/1721.1/72687

David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and comments and can be reached at mazeld at gmail dot com.