Chaos Theory has been an important pursuit of physicists and mathematicians since it was brought to the forefront as a distinct phenomenon in the early 1960s. This early exposition got the attention of the scientific community by—in the best tradition of the scientific method—displaying experimental results that did not comport with theory. Mathematicians had various formulations of the ideas, which then began to converge on chaos as a distinct theoretical framework.
The work of Lorenz  has been credited with showing most clearly that something new was afoot. However other studies came out of the woodwork to show the existence of chaos; many of these were of phenomena observed in the past which had been taken for noise, measurement error, or bad design. And chaos became associated very closely with fractals, which were then discovered in many places, including the art world, for example in the work of Jackson Pollock—who did his work years before the theories were developed.
One of the main intuitive ideas behind chaos is straightforward: the observation of some effect which looks random but is in fact deterministic—and further may be characterized by well-defined, exact processes or equations. This intuitive definition is manifested formally in various ways: sensitive dependence on initial conditions, coverage in the limit of an entire space or subspace, the existence of a dense set of periodic orbits. A large body of theory and practice has grown around these ideas, as manifested by the large numbers of publications that go deep into the math, and publications that describe how to detect and measure chaos in natural and artificial phenomena.
Electronic circuits were seen early on as capable of exhibiting chaotic behavior. The exemplar here is the Chua circuit . More circuits, at varying levels of complexity, have been developed over the years which exhibit some degree of chaos.
J. C. Sprott and W. J. Thio’s new book, Elegant Circuits: Simple Chaotic Oscillators, is a compact guide to many of these circuits, and as well serves as a broad overview of the field of chaos, particularly in electronics. The form of the book is almost a catalog, with introductory explanatory material for each section. Each circuit is described in the form of a schematic, a set of differential equations, simulated results, and physical implementation results. The book is divided into sections by major non-linear component type: diode, transistor, tunnel diode, thyristor, saturating amplifier, analog multiplier, nonlinear inductor, and memristor.
The circuits shown have all been built by the authors, and enough information is provided that most electronics experimenters can build them too. Encouraging such construction is one of the main themes of the book.
And pervading the text is the notion of elegance. Elegance is given a specific definition: low parts count, simple relationships among the component values, and component values in nice round numbers. The circuits typically have only a single active component, plus one or two inductors, capacitors, or resistors. In the section on transistor circuits, for example, almost all circuits use only one transistor.
But a very important part of the book outside of the elegance and of the catalog form is the exposition of the concepts of chaos and the theoretical and practical work that underlies the field.
The Introduction includes a Primer on Chaos, which broadly explains concepts such as sensitivity, mixing and folding, density of periodic points, Lyapunov exponents, and basins of attraction. While it’s an excellent overview, reading other material for a basic grounding in chaos is recommended. Sprott’s previous books on chaos are a good entry point, such as Elegant Chaos: Algebraically Simple Chaotic Flows . Another good introduction is Alligood, Sauer, and Yorke’s Chaos: An Introduction to Dynamical Systems . Examples of other books that delve more formally into the math may be found in texts by Devaney  and Elaydi .
The bibliography in Elegant Circuits is extensive, and emphasizes electronics. Related concepts are sprinkled throughout the book; for some just the italicized term is provided. These are excellent jump-off points to your friend the internet. Jerk circuits, switched capacitor circuits, dripping faucets, ergodic chaotic seas, Poincare sections—these are just a few of the terms and concepts you may wish to explore further.
I was especially drawn (strangely attracted) to the analog multiplier circuits. The differential equations are more amenable to analysis than many of the others which use min, max, or sgn. The Lorenz system, for instance, has approximate analytical solutions, as given for example by Muthukumar et. al., in . In addition, they often map well to physical phenomena, such as the Lorenz equations to weather or the jerk equations to moving bodies. These correspondences also convey a kind of elegance.
With respect to the schematics, I note that each chaotic circuit is shown only in its chaotic form. When based on another circuit, such as the Wien bridge oscillator or Colpitts oscillator, the non-chaotic version of the circuit is not shown. Although the internet can be of service, it would have been helpful to show the exact variant of the original circuit from which the chaotic circuit is derived.
The authors invite the reader to join them on the road to finding interesting chaotic circuits. While they admit applications of chaotic circuits have generally not panned out, like me they also acknowledge that the field then is still wide open, awaiting the discoveries and applications to be found in your imagination. I look forward to seeing EDN articles and Design Ideas that address this challenge.
Elegant Circuits is a good jump-off point for chaotic circuits, one that is accessible and practical. And while the essence of this book is down to earth, it will point your way to the clouds as well.
- Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, 1996.
- Robert Devaney, Chaotic Dynamical Systems, second edition, Westview Press, 2003.
- Saber N. Elaydi, Discrete Chaos, second edition, Chapman and Hall/CRC, 2008.
- Edward N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences 20, pp. 130-141, 1963.
- Matsumoto, A Chaotic Attractor from Chua’s Circuit, IEEE Transactions on Circuits and Systems 31, pp. 1055-1058, 1984.
- Muthukumar et. al., Analytical Solution of A Differential Equation that Predicts the Weather Condition by Lorenz Equations Using Homotopy Perturbation Method, Global Journal of Pure and Applied Mathematics, Volume 13, Number 11, pp. 8065-8074, Research India Publications, 2017.
- Julien C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific, 2010.
Larry Stabile spent a career in software, systems, and electronics. He has a BSEE from MIT.