Chaotic chaos in linked electrical circuits.
But the mathematical -- and now physical -- evidence that such a situation can actually occur is mounting. Mathematicians have pinpointed how certain features in equations, including some of those used to describe physical phenomena such as fluid flow, lead to an extreme kind of unpredictability in the solutions to those equations. Physicists have demonstrated similarly erratic behavior in sets of linked electrical circuits.
"You have a deterministic system, yet you lose experimental replicability," says James C. Alexander of the University of Maryland at College Park. "You're always going to have little errors, and such small changes in initial conditions may lead to completely different long-term behavior."
Alexander described these developments at a Mathematical Association of America meeting held last week in San Francisco.
A wide range of physical systems shows the sensitive dependence on initial conditions that characterizes chaotic behavior. For example, in certain electronic circuits, small changes in the initial voltage can alter the pattern of voltage fluctuations that occur at later times in the circuit.
Nonetheless, a chaotic system's long-term, average behavior remains qualitatively predictable. A particular set of initial values leads to a certain type of outcome, even though one can't predict the details of that behavior.
Equations representing such systems can also have multiple solutions, or attractors, which correspond to different types of behavior. In some cases, a certain set of starting values may lead to one attractor, whereas other values lead to a completely different attractor.
In 1992, Alexander and his colleagues found a theoretical example in which the set of initial conditions, or basin of attraction, leading to one attractor is riddled with points corresponding to initial conditions leading to another outcome. In other words, the two basins are completely intermingled (SN: 11/14/92, p.329).
Soon after, John C. Sommerer of the Johns Hopkins University Applied Physics Laboratory in Laurel, Md., and Edward Ott of the University of Maryland discovered similar behavior in a differential equation representing the motion of a particle traversing a force field having a particular geometry (SN: 9/18/93, p.180). They also suggested that behavior indicative of riddled basins could occur in oscillator circuits that generate chaotic signals. It's possible to link these circuits to synchronize their chaotic behavior (SN: 10/12/91, p.239).
Recently, Peter Ashwin, Jorge Buescu, and Ian Stewart of the Mathematics Institute at the University of Warwick in Coventry, England, investigated the behavior of an identical pair of such circuits. By weakening the link between the two circuits, they showed how the system's behavior gradually changes from a totally synchronous state to one in which the oscillators operate independently, passing through a stage in which the system switches unpredictably in and out of synchrony.
"We studied the details of this loss of synchrony and showed that it is entirely consistent with the riddled-basin scenario," Stewart says.
James F. Heagy, Thomas L. Carroll, and Louis M. Pecora of the Naval Research Laboratory in Washington, D.C., used a system consisting of four linked oscillator circuits. They concentrated on the details of the system's behavior when the connection between the circuits was weak enough to allow both synchronous and nonsynchronous behavior.
"Our work has shown that riddled basins indeed exist in real physical systems . . . and must be taken into account for a proper understanding of the system," the researchers say. They report their findings in the Dec. 26, 1994 Physical Review Letters.
Now, researchers are searching for traces of riddled basins in other settings. But the documented evidence is skimpy. These are not the kinds of results normally reported in journals.
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|Title Annotation:||experimental replicability lost in deterministic systems|
|Date:||Jan 14, 1995|
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