THE BIRTH OF “CHUA’S CIRCUIT”
Many years passed following Lorenz’ work before it was firmly established that chaos can be observed in real-life physical systems. In fact, at first, few paid much attention to Lorenz’ work.
Leon Chua was a faculty member in the Department of Electrical Engineering and Computer Sciences at the Universi-ty of California at Berkeley in 1983. His area of specialization was nonlinear circuits. That year, Chua was awarded a special research fellowship at Japan’s Waseda University.
Upon his arrival there, Chua was invited to witness what was supposed to be the world’s first successful electronic circuit simulation of the Lorenz equations. The research group of a Professor Matsumoto at Waseda University had worked on this project for over a year.
The intended demonstration did not work for one simple reason. The analog multipliers available at the time had neither the nearly-ideal characteristics required nor sufficient dynamic range. Without appropriate analog multipliers, the simulation was not possible.
At that time, only the Lorenz equations and one other nonlinear set called the “Rössler equations” (developed by the German biochemist Otto Rössler in 1976) were known to be chaotic. Compounding the uncertainty that chaos was an actual phenomenon was the fact that chaotic responses had been seen only when computer solutions of these equations were attempted.
Almost all researchers still thought that chaos was merely a phenomenon that could only be observed in computer solutions of special sets of non-linear equations. Chua now hoped to dispel this notion.
CHUA WENT TO WORK.
Leon Chua understood the mechanism that gave rise to chaos in both Lorenz’ and Rössler‘s systems of equations. Im-mediately following his observation of the failed attempt to simulate Lorenz’ equations electronically, Chua decided to design a simpler circuit with these same chaos producing characteristics.
Within a day, Chua designed a circuit and Matsumoto programmed the associated circuit equations on a computer. A strange attractor was produced, confirming the existence of chaos. The circuit that Chua wanted was born!
(Click on the image to enlarge it.)
The circuit that Chua designed could hardly have been simpler. It consists of an inductor (L, with internal resistance, r), two capacitors (C1 and C2), a linear resistor (R), and a negative nonlinear resistor (Rn). Not surprisingly, this circuit soon become known as “Chau’s Circuit.”
All the components for Chua’s Circuit are standard, off-the-shelf items except for the negative non-linear resistor. It is solely the nonlinear resistor that makes Chua’s Circuit nonlinear.
The circuit that Chua designed has been shown to be the simplest autonomous electronic circuit capable of exhibiting chaotic behavior. Click on the link below for an article by Chua et al. which gives more details concerning this circuit.
WHAT IS A NEGATIVE NONLINEAR RESISTOR?
The i vs. v graph of an ordinary resistor is a straight line with a positive slope. In addition to their other functions, these resistors receive electrical energy from the voltage or current sources in a circuit and produce heat. A negative resistor, in contrast, supplies energy to circuit and its i vs. v graph has a negative slope.
The nonlinearity is achieved by having the i vs. v graph consist of several connected straight line segments rather than just one straight line. Negative nonlinear resistors are easily created using op amps (operational amplifiers).
Chua wanted his negative nonlinear resistor to have the shape shown below.
i vs v GRAPH FOR CHUA’S DIODE
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Current is plotted on the vertical axis and is labeled f(v) to denote that it is a function of the voltage across the nonlinear resistor. The slopes of the several straight line segments are represented by m0 , m1 and m2 .
This negative nonlinear resistor came to be known as “Chua’s Diode.” It resulted in chaos with a variety of different strange attractors being produced, depending on the value of the resistor R which connects C1 and C2.
Chua’s Diode has been implemented in several different ways by different authors. The form I prefer is shown below. (Chua’s Diode is the portion to the right of C1.)
CHUA’S CIRCUIT WITH CHAU’S DIODE
(Click on the image to enlarge it.)
Click on the link below to read Chua’s own account of how he developed this circuit.
Zhong and Ayrom experimentally verified in 1985 that Chua’s Circuit does, indeed, produce a chaotic response for certain values of the resistor R that connects C1 and C2. Their paper can be seen by clicking on the link below.
Values for all the components used in Chua’s circuit are given in the three references listed further below.
ENOUGH TALK! LET’S BUILD THE CIRCUIT.
Whether you plan to simulate Chua’s Circuit using LTspice or plan to build it out of actual electronic components, there are several references you will want to read. These articles were written by some of Chua’s former graduate students who are now university faculty members.
The articles explain the design and implementation of Chua’s Circuit very clearly and are tutorial in nature. All component values are listed in these articles.
The references are listed below and can be read and saved in pdf format. Click on the title you wish to read.
The articles assume that you plan to build Chua’s Circuit using electronic components. However, the schematic diagrams and other information are extremely useful even if you only plan to simulate the circuit using LTspice.
These articles, especially the first one, are practically “cookbooks.” They lead you by the hand and explain how the circuit values were chosen.
The first article truly is designed for upper level high school students. The second two articles are a little more advanced but are still very understand-able.
The op-amps (operational amplifiers) used in these circuits are not critical. If you plan to simulate Chua’s Circuit using LTspice, I recommend that you use the LT1113 op-amp which is a very good substitute for the op-amps used in the articles cited above.
Another very good substitute is the LT1057 op-amp. These LT op-amps are available in the component library of LTspice. If you already are an experienced LTspice user, you can add the exact op-amps used in the articles to the LTspice component library, if you wish.
For those planning to construct the actual physical circuit, the only somewhat critical component is the inductor. I recommend that you use one of the following 18 mH inductors:
Bourns components are available from Mouser (www.mouser.com) and from Digi-Key (www.digikey.com).
which is available directly from Coilcraft (www.coilcraft.com)
These inductors have low resistance and are known to work well in Chua’s Circuit.
The op-amps can be practically any general purpose units. There is nothing critical about the chaotic circuits we are and will be discussing.
Either of the Linear Technology op-amps listed above will work well. However, my favorite op-amp when building real circuits is the TL082. This op-amp is very inexpensive and is widely available.
The LMC6482 used in one of the articles is a very good op-amp but is no longer being manufactured. The TL082 likely will be around for a very long time due to its common usage.
While the pin numbering scheme is the same on most op-amps, I am including the TL082 datasheet so that you can check it and be sure.
MY RESULTS WITH CHUA’S CIRCUIT
I have built (and simulated) five or six different versions of Chua’s Circuit using a variety of op-amps and all have worked well from the onset. Remember that the resistor connecting C1 and C2 should be a 2K Ohm potentiometer so that its value can be adjusted easily.
Whether you plan to simulate Chua’s Circuit using LTspice or plan to build it using actual electronic components, you will be able to see many different strange attractors.
So that you can see what you can easily replicate, I have displayed below a few of the many strange attractors I obtained when I simulated Chua’s Circuit using LTspice.
(The reason I am not displaying the strange attractors I saw when I built the circuit using actual electronic components is that I cannot record the displays on my analog oscilloscope.)
(Click on the images to enlarge them.)