  The collective behavior of the ensembles of coupled nonlinear oscillator is
one of the most interesting and important problems in modern nonlinear
dynamics. In this paper, we study rotational dynamics, in particular space-time
structures, in locally coupled identical pendulum-type elements chains that
describe the behavior of phase-locked-loop systems, distributed Josephson
junctions, coupled electrical machines, etc. The control parameters in the
considered chains are: dissipation, coupling strength, and number of elements.
In the system under consideration, the realized modes are synchronous in
frequency and synchronous (in-phase) or asynchronous (out-of-phase) in phase.
In the low dissipation case, the in-phase synchronous rotational regime
instability region boundaries are theoretically found and the bifurcations
leading to the loss of its stability are determined. The analysis was carried
out for chains of arbitrary length. The existence of various out-of-phase
synchronous rotational modes types is revealed: completely asynchronous in
phases and the cluster in-phase synchronization regime. Regularities of
transitions from one type of out-of-phase synchronous mode to another are
established. It was found that at certain coupling parameter values, the
coexistence of stable in-phase and out-of-phase synchronous modes is possible.
It was found that for arbitrary chain length, the number of possible stable
out-of-phase modes is always one less than the chain elements number.
Analytical results are confirmed by numerical simulations.
