D.G. Aronson, M. Golubitsky and M. Krupa
Large arrays of Josephson junctions and iterates of maps with S_n symmetry
Nonlinearity. 4 (1991) 861-902.
Recently models describing the dynamics of large arrays of Josephson junctions coupled through a variety loads have been studied. Since, in applications, these systems are to be operated in a state of stable synchronous oscillation, these studies have emphasized how the synchronous periodic state of stable synchronous oscillation emphasized how the synchronous periodic state can lose stability. A common feature of the models equations is that they are invariant under permutation of the individual junctions. In our study we focus on the effects that these symmetries have on the resulting bifurcations when the synchronous solution loses stability.
In these systems the causes for loss of stability are: fixed-point bifurcations and period-doubling bifurcations. Moreover, these two bifurcations can coalesce in a new codimension-two bifurcation which we call a homoclinic twist bifurcation. Due to the S_n symmetry, it can be shown that the fixed-point bifurcations must lead to families of unstable periodic orbits. The period-doubling bifurcations, however, can lead to stable period-doubling oscillations, and the possible states and their stabilities are classified. In particular, generically, all of the period-doubled oscillations are described by dividing the junctions into two or three groups within which the junctions oscillate synchronously. The existence of these states in the model equations have been confirmed by numerical simulation.
In addition to these period-doubled states, the existence of the homoclinic twist bifurcation and periodic solutions where the junctions oscillate with the same waveform but (math) of a period out of phase with each other is observed in the numberical simulation. These last types of solution are called ponies on a merry-go-round (POMS). In these equations POMS do not arise from a local bifurcation. This issue is discussed in the companion paper.