Circuits That Get Chaos in Sync

JOSEPH NEFF and THOMAS L. CARROLL devised the experiments described here by adapting Carroll's design of the first electronic circuits to exhibit synchronized chaos. Neff, who built the circuits during his senior year at the College of Wooster, recently graduated and is currently pursuing graduate studies in physics at the Georgia Institute of Technology. He programs computers to investigate chaos. Carroll , who received his Ph.D. in 1987 from the University of lllinois, works at the U.S. Naval Research Laboratory. He looks for applications of chaos and has patents pending on some chaotically synchronized circuit designs.

Chaos is not always so chaotic. In some sense, it can be predictable: two systems can be designed so that one exhibits exactly the same chaotic behavior as the other. In other words, the systems would be synchronized. Such devices might be useful for encrypted communications. For example, one of the systems could conceal a message within the chaotic signal. Only someone who possesses the second system would be able to decode the transmission, by subtracting the chaotic signal and leaving behind the message [see "Mastering Chaos," by William L. Ditto and Louis M. Pecora, page 62].

Louis M. Pecora of the U.S. Naval Research Laboratory first came up with the idea of synchronized chaotic systems. He and one of us (Carroll) used a computer simulation to show that such a phenomenon can exist. The next step was to demonstrate the idea using physical systems-specifically, electrical circuits, which are accessible and inexpensive. The first circuit that Carroll built to display synchronized chaos was based on a design devised by Robert Newcomb of the University of Maryland.

Although the circuits described here are simplified versions of Newcomb's, experience with circuit assembly might be helpful. A good introduction to chaos in electrical circuits, using only a diode, an inductor and a resistor, appeared in this column last year [see "How to Generate Chaos at Home," conducted by Douglas Smith; Scientific American, January 1992].

Basically, the setup will consist of two circuits: a driving circuit and a synchronized circuit [see illustration below]. The two are identical except for an important component missing in the synchronized circuit. The two circuits are connected at a single point. The chaotic output of the synchronized circuit will match that generated by the driving circuit if both circuits are correctly built.

Chaotic Circuits produce the same output. They draw current from a 12- volt power supply and are probed by an oscilloscope (left). The values for the components used should match those shown in the schematics (above) to within 1 percent.

Before buying the electronics parts and constructing the two circuits, make sure you have an oscilloscope, breadboards and a power supply that can deliver 12 to 15 volts of direct current. ( Breadboards are thin sheets of plastic that have holes to accommodate electronic components; purchase ones that will hold the positive, negative and ground terminals from the power supply.) The power supply and breadboards are cheap and easy to come by.Oscilloscopes, unfortunately, are expensive; they start at about $500. You can sometimes buy a cheap, used oscilloscope at a "hamfest" -a flea market for ham radio enthusiasts. If you do not wish to purchase an oscilloscope, you might be able to get access to one at a local college laboratory.The device should provide two channels and be able to plot the input to the channels against each other.

For the circuits themselves, you will need various resistors, capacitors and integrated chips called operational amplifiers (op-amps). A list of such items and the minimum quantity you need appears below. We recommend buying more than the minimum, because it is easy to make connection errors that can burn out the components, especially the op-amps.

Because the two circuits must be as similar as possible, it is essential to use resistors and capacitors that have high tolerances. Look for resistors rated to be accurate within 1 percent and capacitors made from polypropylene, which do not leak much current. All but two of the op-amps are generic type 741. The exceptions, labeled "A4" on the schematic [see top illustration on preceding page], are so-called high-frequency uncompensated op-amps. The particular ones used here were type NE5539N. These two are the most critical components of the circuits, as they are ultimately responsible for keeping the circuits synchronized.

The components we used were purchased from Digi-Key Corporation in Thief River Falls, Minn. The total price was less than $30. When ordering, remember to ask for the pin diagrams for the op-amps, which contain several connections that all look alike. The specification sheet describes the configuration of the pins so that you will know which ones to use.

To decrease the possibility of wiring error, it is best to lay out the circuits according to the schematics before connecting them. Start with the op-amps, using one breadboard for each circuit. Observe that the synchronizing circuit is identical to the driving circuit, minus an op-amp and its ancillary components.

The next step is to make the power connections (be sure the power supply is unplugged before beginning). Only the op-amps draw current. Attach wires from the positive, negative and ground terminals to separate rows that run along the top and bottom of the breadboard. These rows are dedicated solely to providing voltage to the electronic components. Use the op-amp spec sheet to determine the appropriate connections from the pins to the rows.The ground wires should be as short as possible because long ground connections can pick up noise, which would keep the circuits from synchronizing.

After hooking up the power supply, begin connecting the components systematically. Although the schematics are simple, it is easy to create a mess. Short, colour-coded wires make the circuit convenient to read and check over. Take several different visual walks through the circuits to verify that all of the connections are correct. Writing down the colour codes of the resistors next to their values on the schematic may speed up this process.

OSCILLOSCOPE PATTERNS typically found in working synchronized circuits are shown above. The chaotic output from one circuit is revealed by a plot of output voltage over time (a); several sweeps are shown. Plotting the output from A2 against that from A3 yields a chaotic attractor (b). Comparing the output of both circuits at X5 shows the circuits are perfectly synchronized (c) or partially so (d).

Once you are satisfied that your circuits are correct, connect the synchronizing circuit to the driving circuit at the points marked "X1" on the schematic. Notice that as a result of the connection, the two circuits have the same nonlinear driving component (the subsystem that uses the A2 op-amp). Connect each channel of the oscilloscope to the points marked "Xs" on the schematic. Each such point is located just after the A1 op-amp. Make sure that the circuits and the oscilloscope all have the same ground - that is, the grounding wire attached to the oscilloscope's probes should be hooked up to the same ground line as the circuit.

Check the circuits several times before plugging in the power. If you burn out one of the op-amps, you can plan to spend at least a few minutes trying to figure out which component is broken and replacing it.
You can verify that the circuits are producing chaos by displaying information from one channel only. If that circuit's output is chaotic, the oscilloscope image will not remain stable on any setting. Instead it produces a pattern similar to a sine wave that rapidly changes in amplitude.

A detailed explanation of why chaos arrives in the driving circuit is rather complex. Briefly, the two integrating op-amp components (the op-amps that have capacitors, labeled "A2" and "A3") are connected in a loop to generate a sine wave. The op-amp components A4 and AS amplify this sine wave, causing its amplitude to increase exponentially over time. Once the signal reaches a certain amplitude, the high-frequency op-amp (A1) brings the amplitude to zero and starts the process all over again. Chaos creeps in during the switching, when A1 resets the circuit.

You can see if your circuits are producing synchronized chaos by plotting the output of one circuit at Xs with that of the other. If the circuits are perfectly synchronized, the oscilloscope will produce a straight line angled at 45 degrees. Because of the switching, the synchronization will probably not be exact. Two broadened, parallel lines may be the result instead.

Do not be disappointed if the circuits fail to synchronize on the first try. Check for faulty connections. The circuits are rather fussy, so any poorly made connections could wreck the synchronicity. A faulty op-amp could be the culprit. You might find that both circuits generate chaos but are not synchronized. There are two possible explanations. One, components that are supposed to be nearly the same may differ too much; for example, two resistors designated as 1,000 ohms may turn out to be 950 and 1,050 ohms. Two, the response time of the high-frequency op-amps may be insufficient, in which case the op-amps should be replaced.

Once the synchronizing chaotic circuits are working properly, you can create the form of message encryption envisioned by Pecora and Carroll. You will need two more op-amps and several large (high-impedance) resistors. With these components, you can easily build two "summing-amplifier" circuits. Such circuits add two signals together and then invert them. These two circuits are connected to the driving and synchronizing circuits [see illustration below]. The message we encoded was a simple sine wave created by a function generator. The sine wave is fed to one of the summing-amplifier circuits, which combines the sine wave with the chaos generated by the driving circuit and inverts the total signal. Amid the chaos, the sine wave is almost impossible to decipher.

Summing Amplifier Circuit (left) marries an input signal with a chaotic signal and then inverts the combination. Two such circuits, when connected to the synchronized chaotic circuits, produce a form of encrypted communication (bottom).

The synchronizing circuit, however, can readily extract the message from the chaos. Because the synchronizing circuit produces the same chaotic pattern as the driving circuit, all one has to do is add the synchronizing circuit's chaotic signal to the driving circuit's total (inverted) signal. What is left is the message. The second summing-amplifier circuit does this job and reinverts the sine wave to its original form.

The summing-amplifier circuits may cause the entire system to behave quite erratically. the image on the oscilloscope may blow up. The problem is that the external devices drain some current and thereby jolt the circuits from one so-called chaotic attractor-a pattern around which the chaotic signals tend to settle-to another.

Large resistors help to alleviate the problem by preventing current drain. We used resistors of 60 kilo-ohms and one mega-ohm, although 50kilo-ohms will probably do. Another way to get the circuit back to a single attractor is to short out an op-amp periodically. Briefly touch a grounded wire to the output of one of the op-amps (preferably the ones with capacitors). This process randomly resets the state of the circuit. The op-amps will not be damaged so long as the shorting is brief. Turning the power supply off and then on again will also work.

To hear the message, you will need to amplify the output. Op-amps do not deliver much current (and the high-impedance resistors do not help matters). A stereo system is a nice solution.Alternatively, a basic electronics handbook will detail how to build simple amplifying circuits to drive a small speaker.
You might try to encode signals more complex than sine waves. You can send audible messages by using a microphone, which converts sound waves into concealable electrical signals. Or you can try sending Morse code: each "click" sends a constant voltage signal. On the other end, you can wire up a small light bulb or a light-emitting diode (known as an LED), which should flash in unison with the clicks if the circuits are working properly.

You can also demonstrate the effects of attractors in chaos by plotting the output of one of the integrating op-amps versus its neighboring op-amp. Such an input produces a spectacular pulsating spiral pattern and is evidence for a single attractor. Feel free to change some of the components to see what happens. Remember, you are trying to create chaos.

CHAOS ON A CHIP. For the first time physicists have shown that well structured chaos can be initiated in a photonic integrated circuit. Furthermore, this represents the first time scientists have been able to study optical chaos at gigahertz rates. The output of a semiconductor laser is normally regular. However, if certain laser parameters are tweaked, such as by modulating the electric current pumping the laser or by feeding back some of the laser’s light from an external mirror, the overall laser output will become chaotic; that is, the laser output will be unpredictable. To make the chaos even more dramatic (and exploitable) Mirvais Yousefi and his colleagues at the Technische Universiteit Eindhoven (in the Netherlands) use paired lasers, lasers built very close to each other on a chip in such a way that each affects the operation of the other. The Eindhoven chip, using the paired-laser mutual-perturbation approach to triggering chaos, is the first to exhibit chaos directly-revealing telltale strange attractors on plots of laser power at one instant versus laser power at a slightly later instant-rather than indirectly through recording laser spectra. Looking ahead to the day when opto-photonic chips are covered with thousands or millions of lasers, the Eindhoven approach could allow troubleshooters to pinpoint the whereabouts of misbehaving lasers---not only that but possibly even exploit localized chaotic effects to their advantage. According to Yousefi ( other possible uses for chip-based chaos will be the business of encryption, tomography, and possibly even in the establishment of multi-tiered logic protocols, those based not on just on the binary logic of 1s and 0s but on the many intensity levels corresponding to the broadband output of the chaotic laser system. (Yousefi et al., Physical Review Letters, 26 January 2007; text at )
THERMAL LOGIC GATES. Information processing in the world's computers is mostly carried out in compact electronic devices, which use the flow of electrons both to carry and control information. There are, however, other potential information carriers, such as photons, which are parcels of light. Indeed a major industry, photonics, has developed around the sending of messages encoded in pulsed light. Heat pulses, or phonons, rippling through a crystal might also become a major carrier, says Baowen Li of the National University of Singapore ( Li, with his colleague Lei Wang, have now shown how circuitry could use heat---energy already present in abundance in electronic devices---to carry and process information. They suggest that thermal transistors (also proposed by Li's group in Applied Physics Letters, 3 April 2006) could be combined into all the type of logic gates---such as OR, AND, NOT, etc.-used in conventional processors and that therefore a thermal computer, one that manipulates heat on the microscopic level, should be possible. Given the fact that a solid state thermal rectifier has been demonstrated experimentally in nanotubes by a group at UC Berkeley (Chang et al., Science, 17 November 2006) only a few years after the theoretical proposal of "thermal diode," the heat analog of an electrical diode which would oblige heat to flow preferentially in one direction (Li et al, Physical Review Letters, 29 October 2004). Li is confident that thermal devices can be successfully realized in the foreseeable future. (Wang and Li, Physical Review Letters, upcoming article)


NONLINEAR CIRCUITS HANDBOOK. Analog Devices Engineering Staff. Analog Devices, Norwood, Mass., 1976.
DYNAMICS: THE GEOMETRY OF BEHAVIOR, Parts 1-4. Ralph H. Abraham and Christopher D. Shaw. Aerial Press, 1985.
INTRODUCTORY ELECTRONICS. Robert E. Simpson. Allyn and Bacon, 1987.
THE ART OF ELECTRONICS. Paul Horowitz and Winfield Hill. Cambridge University Press, 1991.





Chaos Quantum Logic Cosmos Conscious Belief Elect. Art Chem. Maths

Scientific American  Aug 1993 File Info: Created --/--/-- Updated 29/9/2007 Page Address: