IMAGINE tuning in to your local radio station expecting Beethoven but receiving instead a carefully broadcast signal that sounds like static. Being curious, you measure the signal with an oscilloscope 鈥� it is a sequence of waveforms fluctuating seemingly at random. If you were a mathematician you might say these oscillations were chaotic. You鈥檇 be right. But any engineer who decided that such a broadcast was useless would be wrong. Tune in with the appropriate receiver and the music would be crystal clear.
Using the apparent randomness of chaos to communicate anything may sound paradoxical 鈥� after all, according to chaos theory, simple rules sometimes produce disorganised behaviour. But scientists have known for several years that chaos could be used to transmit digital information, in theory at least.
Now scientists in the US have devised a way to do this using the chaotic oscillations of a simple circuit controlled by a computer. Physicists Scott Hayes at the US Army Research Laboratory in Adelphi, Maryland, and Celso Grebogi and Edward Ott of the University of Maryland at College Park are keen to find out whether their method is more efficient, reliable and inexpensive than conventional broadcasting technologies. If it is, cheap, chaotic transmitters could one day replace the expensive circuitry currently used to broadcast signals. More excitingly, but very much in the future, their ideas may also help to explain how the brain encodes and processes information and how this information passes along nerves in the body.
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Although chaotic behaviour may seem impossibly disorganised, behind the apparent randomness lies a complex order. One reason for this complexity is that chaotic processes are extremely sensitive to small disturbances, called perturbations, which can produce large effects. The so-called butterfly effect, for example, describes the idea that the chaotic evolution of the weather can be altered by the atmospheric disturbance produced by the flapping of a butterfly鈥檚 wings. The evolution of the weather may be complex but, in principle, it is also completely predictable if the variables on which it depends are known with infinite precision. In practice, however, any measurement of these variables contains a small error.
The weather, for example, depends on variables such as temperature, wind speed and atmospheric pressure, and these can be measured with limited precision in a few places, giving meteorologists a rough idea of atmospheric conditions. To make a prediction, these measurements are used in a mathematical model of the system which is sensitive to errors in the data. As the calculation proceeds, the errors 鈥� the differences between the real world and the mathematical model 鈥� become amplified, and eventually the forecast is too inaccurate to be of any use. In real life, the measurements made today usually allow accurate forecasts of sunshine in London tomorrow. But today鈥檚 data may not be good enough to predict that a small atmospheric disturbance in the Atlantic will bring rain to the entire country next week.
The weather is a good example of a very complex type of chaos. To understand chaotic communication we can examine a much simpler chaotic system known as a double-scroll oscillator, which occurs in an electronic circuit consisting of a handful of carefully chosen resistors, capacitors and transistors. Under certain conditions, the voltage and current in the circuit fluctuate chaotically. The value of the current and voltage at any time can be measured and plotted as a single point on a graph. The line that joins successive points on this graph traces out a number of loops which, when superimposed, form a pair of finely structured lobes resembling a pair of scrolls when seen from the end. The resulting figure is called an attractor, in this case the double-scroll attractor.
Loop the loop
The shape of the double-scroll attractor is critical. As the current and voltage fluctuate, the dot might trace a number of loops in the left-hand lobe, then wander across the graph and trace a loop on the right-hand lobe, return to the left for a time, and so on. The important point is that at any instant, the dot lies in one lobe or the other. If the lobes are labelled 鈥�1鈥� or 鈥�0鈥�, then the oscillations of the circuit can be represented as a sequence of 0s and 1s. For instance, the sequence 011 corresponds to a loop once around lobe 0, followed by two loops around lobe 1. Rather neatly, this sequence of numbers can form a binary code that can be used to transmit information.
But before this can be done, the output must be controlled. Because the double-scroll circuit is chaotic, small changes in the current or voltage have a subtle effect that only becomes evident in time. For instance, it is possible to create a sufficiently small disturbance the equivalent of the butterfly鈥檚 wings flapping 鈥� which does not affect the first digit or the second or third. But as the system evolves, the disturbance 鈥� increases to the point where numbers after the tenth digit are different from the series that a system without any disturbance would have produced.
By electronically 鈥渘udging鈥� the circuit after each oscillation it is possible to control the future output entirely. In fact, this is the only reliable way to control a chaotic process 鈥� a large disturbance may change the conditions that created the chaotic behaviour in the first place, possibly destroying the attractor and the very oscillations that would otherwise be used to transmit data.
That鈥檚 the theory, at least. In practice, the difficult part is knowing precisely how a disturbance will affect the process. Hayes, Grebogi and Ott tackled the problem by applying a perturbation to the circuit and using a computer to predict the effect ten digits in advance. By repeating this process after each oscillation, they were able to compile a database that showed them how to control the tenth digit given any preceding combination of digits. Armed with such a database, they could control the output of the double-scroll oscillator after it generated the first nine digits.
Obviously, controlling a signal is the key to using it for communication. A simple 鈥渃haotic鈥� transmitter could be built using a device which produces microwaves and behaves chaotically at high power. To broadcast a message, the current and voltage in the tube would be measured by a computer and used to predict the output over the next nine oscillations 鈥� the idea being that the tenth, which can be controlled, forms the first digit in the message.
By consulting the database, the computer can work out how to nudge the input power so that the tube produces the desired output during the tenth oscillation. After the first nudge, the computer calculates how to control the eleventh oscillation, which forms the second digit in the message, and so on. The process can be repeated indefinitely to send a message of any length. The receiver need only discard the first nine bits of the signal to retrieve the message.
Order out of chaos
But why would this way of communicating be better than existing methods? There seem to be several advantages. For a start, conventional methods of digital communication rely on amplifiers to boost the tiny signal from a microprocessor to a powerful one that can be broadcast over long distances. Simple amplifiers that can handle periodic signals at low power are relatively inexpensive to build. But if the power is turned up, these simple amplifiers tend to behave chaotically, so conventional transmitters contain complex and expensive circuitry to steady the signal. Chaotic transmitters do not need this additional circuitry, however. The big advantage of chaotic communication is that there is no need to amplify the message, says Hayes, only to convert it into a sequence of small fluctuations in voltage that can steer the output of the tube which is already operating chaotically at high power.
Chaotic broadcasts may also be less susceptible to the errors caused by background noise or interference. At the moment, data must be encoded in such a way that if any are lost or corrupted, the entire message can be reconstructed. Telecommunications companies use this kind of transmission for international telephone calls relayed through a satellite. The encoding schemes are complex and involve a computer that encodes the data before it is sent and another to decode it on arrival.
A chaotic system, on the other hand, has a built-in error checking capability. Although the oscillations of a chaotic process appear erratic, they are not random and certain patterns crop up again and again. For instance, the double-scroll circuit never traverses the lobe O twice in a row. (Messages that contain consecutive 0s can be sent by replacing every 0 in the message by 01.)
Put another way, the sequence of digits generated by chaotic oscillations have a kind of grammar: some sequences of 0s and Is are allowed and others are not. The rules used in human languages are a good analogy, says Ott. 鈥淥nly certain combinations of letters can form words, and grammar imposes rules on how these words can be strung together. This allows you to spot errors,鈥� he explains. In the case of the double-scroll oscillator, the receiver knows that the transmission has been corrupted if two consecutive 0s appear in the signal in the same way that English speakers know that something is wrong if two qs appear consecutively in a word.
The grammar of chaos provides a context that can be used to identify and correct errors that occur during the transmission 鈥� much as we can replace a garbled or missing word in a sentence by analysing the context. This property of a language, be it chaos or English, is known as redundancy. Hayes is currently trying to find out whether chaos provides better redundancy than conventional transmission methods.
On a philosophical note, the work at Maryland gives a curious insight into the way mathematical ideas relate to the real world. Mathematicians frequently borrow ideas from one field to help them describe concepts in another. When chaos theory was developed some of the ideas needed to explain it were borrowed from a branch of mathematics dealing with information and how to transmit it. At the time, scientists had no idea that chaos could be used to communicate. 鈥淭his is one of those interesting, fundamental connections between an idea in mathematics and something that has potential applications in the real world,鈥� says Ott.
Chaos may also play an important part in the way the brain holds information and the way it passes through the nervous system. 鈥淟ittle is known about how most of the brain encodes and transmits information,鈥� says Steven Schiff, a neurosurgeon at the Children鈥檚 National Medical Center in Washington DC. 鈥淥ne of the central problems confronting neuroscientists is to crack this neural code,鈥� he explains. One intriguing possibility is that the nervous system may use chaos to encode and transmit data. Paul Rapp, a neurophysiologist at the Medical College of Pennsylvania, says that mathematical models of the way neurons work can exhibit chaotic behaviour. But he points out that nobody has actually observed chaos in real neurons.
Schiff is one of the researchers who is looking for this chaotic behaviour. His experiments seem to show that the electrical signals in tissue in the spinal cord and in the brain sometimes exhibit patterns that are similar to those of chaotic attractors, such as the double-scroll oscillator, but as yet he has no proof that they are chaotic. While this is a step in the right direction, Schiff believes it is a long way from cracking the neural code. Should anyone succeed, however, the rewards could be great. An understanding of the neural code may one day allow scientists to build prosthetic devices that can interpret and carry out complex commands from the brain. For example, the right type of signals generated outside the body and fed to paralysed limbs could help restore their lost function.
These devices are a long way from being realised, of course. But the feasibility of communicating with chaos has already been demonstrated by physicists. It may not be long before the hissing static that we now avoid actually transmits the majestic tones of Beethoven.