A four-level model which takes account of the polarization of the laser field by including the spin sublevels of the conduction and valence bands of a semiconductor allows us to introduce vector rate equations which account for the polarization degree of freedom of the laser emission. Analysis of these rate equations and their extension to include transverse degrees of freedom provides important physical insight into the nature of polarization instabilities in surface-emitting semiconductor lasers. In the absence of transverse effects the model predicts a marginally stable linearly polarized state. The type of dynamical response of the polarization degrees of freedom is linked to the relative time scale of spontaneous-emission and spin-relaxation processes. With transverse effects included, we predict the existence of stable transverse spatially homogeneous intensity outputs with arbitrary direction of linear polarization in the transverse plane. The stability of the off-axis emission solutions to long-wavelength perturbations is investigated and, in addition to an Eckhaus instability associated with a global phase, we predict a polarization instability associated with a relative phase of the complex field vector. The role of phase anisotropy in the laser cavity is explored close to threshold and we predict that it stabilizes two preferred orthogonal directions of polarization, which, however, are discriminated in their stability properties by transverse effects.

We consider stochastic differential equations for a variable $q$ with multiplicative white and nonwhite ("colored") noise appropriate for the description of nonequilibrium systems which experience fluctuations which are not "self-originating." We discuss a numerical algorithm for the simulation of these equations, as well as an alternative analytical treatment. In particular, we derive approximate Fokker-Planck equations for the probability density of the process by an analysis of an expansion in powers of the correlation time $\tau$ of the noise. We also discuss the stationary solution of these equations. We have applied our numerical and analytical methods to the "Stratonovich model" often used in the literature to study nonequilibrium systems. The numerical analysis corroborates the analytical predictions for the time-independent properties. We show that for large noise intensity $D$ the stationary distribution develops a peak for increasing $\tau$ that becomes dominant in the large-$\tau$ limit. The correlation time of the process in the steady state has been analyzed numerically. We find a "slowing down" in the sense that the correlation time increases as a function of both $D$ and $\tau$.. This result shows the incorrectness of an earlier analysis of Stratonovich.

We explore the coupled dynamics of the internal states of a set of interacting elements and the network of interactions among them. Interactions are modeled by a spatial game and the network of interaction links evolves adapting to the outcome of the game. As an example, we consider a model of cooperation in which the adaptation is shown to facilitate the formation of a hierarchical interaction network that sustains a highly cooperative stationary state. The resulting network has the characteristics of a small world network when a mechanism of local neighbor selection is introduced in the adaptive network dynamics. The highly connected nodes in the hierarchical structure of the network play a leading role in the stability of the network. Perturbations acting on the state of these special nodes trigger global avalanches leading to complete network reorganization.

Polarization-state selection, polarization-state dynamics, and polarization switching of a quantum-well vertical-cavity surface-emitting laser (VCSEL) for the lowest order transverse spatial mode of the laser is explored using a recently developed model that incorporates material birefringence, the saturable dispersion characteristic of semiconductor physics, and the sensitivity of the transitions in the material to the vector character of the electric field amplitude. Three features contribute to the observed linearly polarized states of emission: linear birefringence, linear gain or loss anisotropies, and an intermediate relaxation rate for imbalances in the populations of the magnetic sublevels. In the absence of either birefringence or saturable dispersion, the gain or loss anisotropies dictate stability for the linearly polarized mode with higher net gain; hence, switching is only possible if the relative strength of the net gain for the two modes is reversed. When birefringence and saturable dispersion are both present, there are possibilities of bistability, monostability, and dynamical instability, including switching by destabilization of the mode with the higher gain to loss ratio in favor of the weaker mode. We compare our analytical and numerical results with recent experimental results on bistability and switchings caused by changes in the injection current and changes in the intensity of an injected optical signal.

Studies of cultural differentiation have shown that social mechanisms that normally lead to cultural convergence—homophily and influence—can also explain how distinct cultural groups can form. However, this emergent cultural diversity has proven to be unstable in the face of cultural drift—small errors or innovations that allow cultures to change from within. The authors develop a model of cultural differentiation that combines the traditional mechanisms of homophily and influence with a third mechanism of network homophily, in which network structure co-evolves with cultural interaction. Results show that in certain regions of the parameter space, these co-evolutionary dynamics can lead to patterns of cultural diversity that are stable in the presence of cultural drift. The authors address the implications of these findings for understanding the stability of cultural diversity in the face of increasing technological trends toward globalization.

The increasing integration of technology into our lives has created unprecedented volumes of data on society’s everyday behaviour. Such data opens up exciting new opportunities to work...