D. Aronson, M. Golubitsky and M. Krupa
Coupled arrays of Josephson junctions and bifurcation
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.