# Focus: Generating Chaos in a New Way

Published July 1, 2011  |  Phys. Rev. Focus 28, 1 (2011)  |  DOI: 10.1103/PhysRevFocus.28.1

#### Nondeterminism in the Limit of Nonsmooth Dynamics

Mike R. Jeffrey

Published June 24, 2011

Small changes in a system’s conditions–a tiny variation in its temperature, for example–normally have correspondingly small consequences. But abrupt transformations, as when an electrical switch flips or a gene triggers events in a cell, can have surprising consequences, according to the 24 June Physical Review Letters. A theorist argues that such a sudden transition can lead to an “explosion” of unpredictable, chaotic behavior that isn’t apparent from the equations describing the system, unlike many well-studied chaotic systems. He also speculates that this explosive unpredictability may be an unrecognized cause of puzzling behavior in many real systems.

In modelling everything from electrical circuits to global climate, scientists and engineers rely on differential equations that describe how the rate of change of key variables–pressure, voltage, or whatever–depend on the current values of those same variables. Such equations generally involve continuous (smoothly varying) functions, reflecting the fact that small changes in the system’s current state cause equally small changes in its behavior.

But discontinuities can enter such equations if physical properties change abruptly when a variable reaches a certain value. A superconductor, for example, suddenly becomes an ordinary metal if its temperature rises above a critical temperature, ${T}_{c},$ when its electrical resistance suddenly jumps up from zero. The equations describing the system take one form below that temperature and another above.

To analyze the effects of such discontinuities, Mike Jeffrey of the University of Bristol in England used a branch of math called dynamical systems theory. The most dramatic outcomes, his analysis suggests, should appear in so-called “grazing bifurcations,” in which a system approaches a discontinuity but doesn’t necessarily go all the way through it. In the superconductor example, this would mean approaching ${T}_{c}.$

The consequence of such flirtations, he finds, is an “explosion” of unpredictability. On reaching the discontinuity, the system passes through a few short moments in which the forces driving it become indefinite–the system is beyond the realm of either set of equations. It can be kicked one way or another by microscopic fluctuations, which normally have little effect, and its future becomes practically unpredictable. This behavior differs from more conventional chaotic systems, like a pair of weakly coupled pendulums, where a set of continuous equations predicts the system’s future state, but that final state can vary dramatically with only tiny changes in the system’s starting point.

Applying the abstract results to the equations of a real system, Jeffrey shows that such explosions can happen in a ring of the superconductor niobium nitride, which can be used as a highly sensitive magnetic field detector. The electric current in this ring and its temperature change according to two simple differential equations that change abruptly at ${T}_{c}.$ As Jeffrey shows, at ${T}_{c}$ these equations can effectively lose control, with the ring’s temperature and current becoming wildly sensitive to microscopic, molecular-scale fluctuations. A system that might appear to be fully predictable from equations instead flips randomly between two different kinds of behavior, with periods of regular oscillations interrupted sporadically by periods of stasis.

Jeffrey speculates that such non-deterministic chaos, as he calls it, could be common. “We may only have a few current examples of these explosions simply because people haven’t taken them seriously before,” he says. Moreover, the phenomenon might be ignored by an experimenter who might simply turn off a device that has suddenly become highly unpredictable. “The surprise of the analysis is that such points of indecision can be accessed by systems very easily, whereas conventional wisdom holds that they should almost never be seen.”

“This work describes a simple scenario where determinism fails–not just fails slightly, but spectacularly,” says Alan Champneys, of the University of Bristol, who was not involved in the work. Vincent Acary of the National Institute for Research in Computer Science and Control (INRIA) in France says the work is part of a larger trend. “There is a growing realization within the engineering, physical, biological, medical, and social sciences that many nonlinear behaviors can be explained by discontinuities. This article is a good example of some of the surprises that emerge.”

–Mark Buchanan

Mark Buchanan is a freelance science writer in Cambridge, England.