Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

In the last decade, non-linear dynamical transport in semiconductor superlattices (SLs) has witnessed significant progress in theoretical descriptions as well as in experimentally observed non-linear phenomena. However, until now, a clear distinction between non-linear transport in strongly and weakly coupled SLs was missing, although it is necessary to provide a detailed description of the observed phenomena. In this review, strongly coupled SLs are described by spatially continuous equations and display self-sustained current oscillations due to the periodic motion of a charge dipole as in the Gunn effect for bulk semiconductors. In contrast, weakly coupled SLs have to be described by spatially discrete equations. Therefore, weakly coupled SLs exhibit a more complex dynamical behaviour than strongly coupled ones, which includes the formation of stationary electric field domains, pinning or propagation of domain walls consisting of a charge monopole, switching between stationary domains, self-sustained current oscillations due to the recycling motion of a charge monopole and chaos. This review summarizes the existing theories and the experimentally observed non-linear phenomena for both types of semiconductor SLs.

Tunable oscillatory modes of electric-field domains in doped semiconductor superlattices are reported. The experimental investigations demonstrate the realization of tunable, GHz frequencies in GaAs-AlAs superlattices covering the temperature region from 5 to 300 K. The orgin of the tunable oscillatory modes is determined using an analytical and a numerical modeling of the dynamics of domain formation. Three different oscillatory modes are found. Their presence depends on the actual shape of the drift velocity curve, the doping density, the boundary condition, and the length of the superlattice. For most bias regions, the self-sustained oscillations are due to the formation, motion, and recycling of the domain boundary inside the superlattice. For some biases, the strengths of the low- and high-field domain change periodically in time with the domain boundary being pinned within a few quantum wells. The dependency of the frequency on the coupling leads to the prediction of a different type of tunable GHz oscillator based on semiconductor superlattices.

Active Ornstein-Uhlenbeck particles (AOUPs) are overdamped particles in an interaction potential subject to external Ornstein-Uhlenbeck noises. They can be transformed into a system of underdamped particles under additional velocity dependent forces and subject to white noise forces. There has been some discussion in the literature on whether AOUPs can be in equilibrium for particular interaction potentials and how far from equilibrium they are in the limit of small persistence time. By using a theorem on the time reversed form of the AOUP Langevin-Ito equations, I prove that they have an equilibrium probability density invariant under time reversal if and only if their smooth interaction potential has zero third derivatives. In the limit of small persistence Ornstein-Uhlenbeck time $\tau$, a Chapman-Enskog expansion of the Fokker-Planck equation shows that the probability density has a local equilibrium solution in the particle momenta modulated by a reduced probability density that varies slowly with the position. The reduced probability density satisfies a continuity equation in which the probability current has an asymptotic expansion in powers of $\tau$. Keeping up to $O\left(\tau \right)$ terms, this equation is a diffusion equation, which has an equilibrium stationary solution with zero current. However, $O\left({\tau }^2\right)$ terms contain fifth- and sixth-order spatial derivatives and the continuity equation no longer has a zero current stationary solution. The expansion of the overall stationary solution now contains odd terms in the momenta, which clearly shows that it is not an equilibrium.

A discrete model is introduced to account for the time-periodic oscillations of the photocurrent in a superlattice observed by Kwok et al. in an undoped 40-period AlAs/GaAs superlattice. The basic ingredients are an effective negative-differential resistance due to the sequential resonant tunneling of the photoexcited carriers through the potential barriers, and a rate equation for the holes that incorporates photogeneration and recombination. The photoexciting laser acts as a damping factor ending the oscillations when its power is large enough. The model explains (i) the known oscillatory static I-V characteristic curve through the formation of a domain wall connecting high- and low-electric-field domains, and (ii) the photocurrent and photoluminescence time-dependent oscillations after the domain wall is formed. In our model, they arise from the combined motion of the wall and the shift of the values of the electric field at the domains. Up to a certain value of the photoexcitation, the nonuniform field profile with two domains turns out to be metastable: after the photocurrent oscillations have ceased, the field profile slowly relaxes toward the uniform stationary solution (which is reached on a much longer time scale). Multiple stability of stationary states and hysteresis are also found. An interpretation of the oscillations in the photoluminescence spectrum is also given.