  History-dependence and ageing are studied in the low-temperature glass phase
of a long-range periodic Josephson array. This model is characterized by two
parameters, the number of wires ($2N$) and the flux per unit strip ($\alpha$);
in the limit $N \to \infty$ and fixed $\alpha \ll 1$ the dynamics of the model
are described by the set of coupled integral equations, which coincide with
those for the $p=4$ disordered spherical model. Below the glass transition we
have solved these equations numerically in a number of different regimes. We
observe power-law ageing after a fast quench with an exponent that decreases
rapidly with temperature. After slow cooling to a not-too-low temperature, we
see ageing characterized by the appearance of a new time scale which has a
power law dependence on the cooling rate. By contrast if the array is cooled
slowly to very low temperatures the ageing disappears. The physical
consequences of these results in different cooling regimes are discussed for
future experiment. We also study the structure of the phase space in the low
temperature glassy regime. Analytically we expect an exponential number of
metastable states just below the glass transition temperature with vanishing
mutual overlap, and numerical results indicate that this scenario remains valid
down to zero temperature. Thus in this array there is no further subdivision of
metastable states. We also investigate the probability to evolve to different
states given a starting overlap, and our results suggest a broad distribution
of barriers.
