The Demystification of Chaos Through the Chua Circuit – Georgios Ardavanis (Ph.D.)

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We describe far too many situations in our day-to-day lives as chaotic or ask, “What chaos is this?” Take New York’s traffic, for instance. However, we also give disorder a deeper significance. In combination with the natural and environmental sciences, that is. For instance, when discussing how it is impossible to forecast weather for some time into the future and beyond.

Chaos is defined by scientists as the magnified impacts that even minute changes can have in the here and now, and which can eventually result in unpredictability. According to Newton, the cosmos is made up of a set of rules governed by physics. Rules that, when followed, will inevitably produce a particular outcome. On the other hand, chaos theory can demonstrate rather diverse results. For a variety of reasons, even the tightest regulations might have unexpected outcomes. According to chaos theory, complex, chaotic systems have fundamental patterns, self-organization, interconnection, self-similarity, and continuous feedback loops despite their outward appearance of randomness.

Chaos was regarded as a mathematical anomaly of a class of systems that showed extraordinary sensitivity to slight changes in their initial circumstances (the well-known butterfly effect) and were formally characterized by nonlinear differential equations until the early 1980s. The system of three first-order ordinary differential equations published in the 1960s by American mathematician and meteorologist Edward Norton Lorenz as a simplified mathematical model of atmospheric transport is a typical example. Lorenz was a professor at MIT. Nevertheless, until then, the only methods available for studying the Lorenz system and other related chaotic systems were computer simulations conducted in a sterile environment. Consequently, the necessity for experimental validation of chaos emerged in the early 1980s.

One of the initial attempts was carried out in 1983 at Waseda University in Japan by a research team led by Professor Matsumoto. This group created a model of the Lorenz system by designing and building an electrical circuit. The failure of this endeavor stemmed from the absence of a crucial element (the proportional multiplier) required to precisely replicate the nonlinear terms of Lorenz’s system, which consisted of two straightforward products. Nonetheless, this setback encouraged University of Berkeley professor Leon Chua to move forward in 1983 with the creation of an electronic circuit that exhibits chaos without necessarily modeling any known dynamic system from the literature. Remembering that three 1st order ordinary differential equations must mathematically describe an autonomous dynamical system in order for it to exhibit Chaos, Professor Leon Chua designed his circuit with three electrical energy storage elements—two capacitors and an inductor—a linear resistor, a non-linear resistor, and a piecewise linear current-voltage characteristic. As a result, Professor Leon Chua was able to create the circuit design that is now known as Chua’s circuit, which was discovered to really display the intended phasic behavior upon implementation.

This discovery dispelled the mystery around chaos by demonstrating that it is a natural feature of the universe. Consequently, a flurry of research was conducted to find evidence of chaos in systems originating from disciplines like physics, chemistry, and biology. Additionally, Chua’s circuit’s straightforward design made Chaos available to the technical community, who started looking for it as well. Furthermore, because of the Chua circuit’s complex dynamic behavior, researchers can still use it to look for and find Chaos theory-related phenomena. Ultimately, Chua’s circuit was put to use in the creation of neural networks, cryptography, secure communication systems, random number generators, and even the arts, like music.


Georgios Ardavanis – 17/10/2023

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