ON CHAINS OF OSCILLATORS FORCED AT ONE END

Chains of oscillators periodically forced at one end may be entrained to a range of forcing frequencies. That range depends on the nature of the coupling, the size of the chain, and the end that is being forced, and gives clues to the properties of the connections along the chain. For long chains of equal oscillators with coupling in both directions, the unforced chain generically has a pattern of phase lags that depends mainly on the coupling in only one of the two directions. For a large class of coupling functions, entrainment is possible at each end over a range of frequencies that contain an open interval independent of the number of oscillators. For this class, the dominant direction can be deduced from the relationship between the ensemble frequency of the unforced chain and the boundaries of the ranges of entrainment when a long chain is forced at each end. This theory is applied to data from the central pattern generator for undulatory locomotion in a primitive vertebrate. The mathematical technique includes the use of a one-dimensional, locally defined, discrete dynamical system to analyze how the phase relationships among the oscillators change with the forcing frequency and the end being forced. This dynamical system makes possible the analysis of the behavior for long chains. The stability of the solutions is verified using techniques related to monotone methods of partial differential equations. For another class of coupling functions, the techniques show that, even if the coupling in the two directions are of roughly comparable amplitude, the range of entrainment for forcing at one of the ends may shrink to a point as the number of oscillators grows, while the range for forcing at the other end remains bounded away from zero in size.