We explore the effect of noise on the ballistic graphene-based small Josephson junctions in the framework of the resistively and capacitively shunted model. We use the non-sinusoidal current–phase relation specific for graphene layers partially covered by superconducting electrodes. The noise induced escapes from the metastable states, when the external bias current is ramped, given the switching current distribution, i.e. the probability distribution of the passages to finite voltage from the superconducting state as a function of the bias current, that is the information more promptly available in the experiments. We consider a noise source that is a mixture of two different types of processes: a Gaussian contribution to simulate an uncorrelated ordinary thermal bath, and non-Gaussian, α -stable (or Lévy) term, generally associated to non-equilibrium transport phenomena. We find that the analysis of the switching current distribution makes it possible to efficiently detect a non-Gaussian noise component in a Gaussian background.

We propose a threshold detector for Lévy-distributed fluctuations based on a Josephson junction. The Lévy-noise current added to a linearly ramped bias current results in clear changes in the distribution of switching currents out of the zero-voltage state of the junction. We observe that the analysis of the cumulative distribution function of the switching currents supplies information on both the characteristics’ shape parameter $\alpha$ of the Lévy statistics. Moreover, we discuss a theoretical model, which allows characteristic features of the Lévy fluctuations to be extracted from a measured distribution of switching currents. In view of these results, this system can effectively find an application as a detector for a Lévy signal embedded in a noisy background.

We propose a modification of the Kuramoto model to account for the effective change in the coupling constant among the oscillators, as suggested by some experiments on Josephson junction, laser arrays, and mechanical systems, where the active elements are turned on one by one. The resulting model is analytically tractable and predicts that both first and second order phase transitions are possible, depending upon the value of a new parameter that tunes the coupling among the oscillators. Numerical simulations of the model are in accordance with the analytical estimates, and in qualitative agreement with the behavior of Josephson junctions coupled via a cavity.

A model for a large number of Josephson junctions coupled to a cavity is presented. The system displays synchronization behavior very similar to that reported in recent experiments [P. Barbara et al., Phys. Rev. Lett. 82, 1963 (1999)]. The essential dynamical mechanism responsible for coherence should be generic in nonlinear oscillator systems where the interactions are mediated by a highly resonant cavity, in analogy with gas lasers.

Application of the classic McLaughlin-Scott soliton perturbation theory to a Josephson-junction fluxon subjected to a microwave field that interacts with the fluxon only at the junction boundaries reduces the problem of phase locking of the fluxon oscillation to the study of a two-dimensional functional map. Phase-locked states correspond to fixed points of the map. For junctions of in-line geometry, the existence and stability of such fixed points can be studied analytically. Study of overlap-geometry junctions requires the numerical inversion of a functional equation, but the results are qualitatively very similar. The map predicts significantly different behaviors for locking at odd and even subharmonic frequencies and at superharmonic frequencies. It also gives indications regarding hysteresis in the current-voltage characteristic, the existence of zero-crossing steps, and a description of the locking process in the frequency domain. The principal merit of the map is that it captures much of the experimental phenomenology at a very low computational cost.