New Work Sheds Light on Controlling Chaos
The research concerns the science of deterministic chaos, which states that even though the fluctuating behavior of a given system may seem random and unpredictable, these fluctuations are, in fact, part of an overall pattern that can be analyzed and understood. Indeed, several years ago scientists showed that chaotic systems could be controlled, that is, made non-chaotic, by manipulating certain variables relating to the particular system. This idea can be illustrated with the simple analogy of a circus performer trying to balance a broomstick in the palm of his hand. The broomstick is very unstable. Sometimes it falls down immediately, and sometimes it might stay upright for a second or more. At any rate, it is very difficult to predict when the broomstick will fall, said Ernst Niebur, an assistant professor of neuroscience at The Johns Hopkins University's Zanvyl Krieger Mind/Brain Institute.
The system is said to be chaotic; its behavior is seemingly random and unpredictable and depends on initial conditions that are exceedingly difficult to duplicate. The performer can never put the broomstick in exactly the same position in his palm two times in a row. Minute differences are immensely magnified by the system, so that it will behave differently every time. Nevertheless, the system is deterministic: the chaotic behavior is not random at all but is directly related to real, pre-existing conditions. For all practical purposes, however, scientists do not know the conditions in enough detail to predict the behavior with any precision.
Perhaps the most obvious example of a chaotic system is the weather: it can be profoundly influenced by minute variables, making prediction a constant challenge.
But researchers discovered several years ago that it is possible to make minute changes to the properties of a chaotic system, thereby controlling the chaotic behavior.
"The broomstick is a good example of how you can control the chaos," Niebur said. "You are capable of balancing the broomstick by moving your hand around, adjusting the parameters."
In the broomstick-balancing analogy, the parameters correspond to adjustments of motion and position. In other systems, the parameters might correspond to adjustments of energy, stimuli or chemical agents.
Niebur and Heinz Schuster, a professor of theoretical physics at the University of Kiel in Germany, have now devised equations to pinpoint the controlling mechanism. They discovered that a chaotic system can only be controlled if the parameters to be changed fall within a specific range of values. If the computations are applied to a two-dimensional system, a system in which there are only two types of parameters, the results are striking: when the system is plotted on a graph, control can be achieved only if the values of the parameters fall within a triangular region on the graph. For parameters that fall outside of this triangle, adjusting them will not result in control and the system cannot be stabilized.
For systems in more than two dimensions, the range of parameters gets more complicated; for example, in a three-dimensional system, the parameters must fall within a somewhat twisted pyramidal shape. For higher dimensions, human intuition cannot imagine the high-dimensional shape of the control region but it is possible to compute it, Niebur said.
Schuster and Niebur submitted a paper on their findings to the journal Physical Review Letters. Scientists reviewing the paper, however, said they needed to be convinced: the calculations looked fine, but without hard experimental evidence, they just weren't sold on the concept. The theory was just that, a theory.
Niebur and Schuster then sought help from physicist Earle Hunt and his colleagues at Ohio University in Athens. They tested the theory on a simple electronic circuit. Hunt had shown several years ago that in such a circuit, as the incoming voltage is increased, the current inside the circuit fluctuates wildly, exhibiting chaos. Furthermore, he found that he could stabilize the current by adjusting two parameters in the circuit -- essentially, by varying the degree of electrical resistance. When the Ohio University physicists heard about the work by Schuster and Niebur, they looked more closely at the range of parameters in which they could stabilize the system.
Sure enough, they confirmed that control was possible only within a triangular region on a graph of the system.
"Within a week or so they sent us back a fax with this triangle," Niebur said. The reviewers were satisfied. "After they saw the experimental confirmation, the paper was immediately accepted."
"It is not really important whether it's a circuit or whether it's a mechanical system or even the brain," Niebur said. The mathematical theory may be applicable to all of those systems, and the most important result of the experiment is that it proved this theory correct, he noted.
The scientific paper, published Jan. 15 in Physical Review Letters, was written by Niebur, Schuster, and their Ohio University collaborators.
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