History dependence and aging in a periodic long-range Josephson array

History dependence and aging 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 $\mathrm{}\left(2N\right)\mathrm{}$ and the flux per unit strip $\left(\alpha \right)$; 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 $p=4\mathrm{}$ disordered spherical model. Below the glass transition we have solved these equations numerically in a number of different regimes. We observe power-law aging after a fast quench with an exponent that decreases rapidly with temperature. After slow cooling to a not-too-low temperature, we see aging characterized by the appearance of a 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 aging 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.