Fireflies blink synchronously, lasers are “mode-locked” for amplification, and cardiac pacemaker cells maintain a steady heartbeat. In each case, the interaction of separate oscillators leads to important collective behavior. Although physicists have been investigating so-called coupled oscillators for centuries, there has been relatively little study of the effects of time delays, which always occur to some extent as one oscillator communicates with another. In the 16 October PRL a team shows that a very simple electronic circuit containing two identical oscillators coupled with a time delay can experience “amplitude death”–the oscillators essentially stop each other cold. The authors also found several other new phenomena resulting from the delay, all of which could be applied in areas from sophisticated laser electronics design to research on heart attacks.
Ever since the Dutch physicist Christiaan Huygens noticed in 1665 that two pendulum clocks in his room tended to swing in synch, researchers have been fascinated by coupled oscillators. Physicists and mathematicians have now learned that, depending on the strength of the coupling between two oscillators and the difference between their natural frequencies, a range of phenomena can occur: synchronous oscillation, incoherent oscillation, or amplitude death. But no one had seen amplitude death for a pair of identical-frequency oscillators until Abhijit Sen and his colleagues at the Institute for Plasma Research in Bhat, India, predicted it theoretically for oscillators coupled through a delay . When amplitude death popped out of the equations, “that was a big surprise,” recalls Sen. Many real-world coupled oscillators–like fireflies and pacemaker cells–have nearly identical frequencies, so their predictions seemed significant.
Now Sen and his colleagues have coupled two so-called LCR circuits–which oscillate indefinitely thanks to nonlinear resistors–through a digital delay and verified their predictions, while observing additional behaviors as well. Just as they had predicted, the team found several ranges of coupling strengths and time delays that caused the oscillators to poop out. Each range was distinct from the others, so the researchers call them “death islands”–a feature not seen in systems without a delay. They also observed “antiphase” oscillations, which are the electronic equivalent of two pendulums swinging alternately toward and away from one another. Normally only the in-phase or the antiphase mode can exist in this type of nonlinear oscillator, but not both. Sen points out that such flatlining or antiphase behavior could be destructive for a set of coupled lasers, so understanding these phenomena may help electrical engineers.
Paul Linsay, formerly of MIT, cautions that all of these effects pertain to “limit-cycle” oscillators–those that always tend toward a specific frequency and amplitude–and not to simple harmonic oscillators. He says that coupled oscillator equations with time delays are usually “murderously hard to solve,” so delay effects have been largely ignored in the past. Linsay thinks the new results could be applicable to coupled systems of periodically firing neurons, such as those in the brain. This work suggests that a set of firing neurons that represent a memory might suddenly stop. “Maybe that’s why you forget things for a while,” he speculates, “then you remember them two days later.”
- D. V. Ramana Reddy and A. Sen, and G. L. Johnston Phys. Rev. Lett. 80, 5109 (1998).