We introduce a general methodology of update rules accounting for arbitrary interevent time (IET) distributions in simulations of interacting agents. We consider in particular update rules that depend on the state of the agent, so that the update becomes part of the dynamical model. As an illustration we consider the voter model in fully connected, random, and scale-free networks with an activation probability inversely proportional to the time since the last action, where an action can be an update attempt (an exogenous update) or a change of state (an endogenous update). We find that in the thermodynamic limit, at variance with standard updates and the exogenous update, the system orders slowly for the endogenous update. The approach to the absorbing state is characterized by a power-law decay of the density of interfaces, observing that the mean time to reach the absorbing state might be not well defined. The IET distributions resulting from both update schemes show power-law tails.

The dynamics of patterns in large two-dimensional domains remains a challenge in nonequilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full two-dimensional generalizations of the latter can lead to unexpected dynamic behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, which is a generic model of anisotropic pattern forming systems and has been derived in different instances of thin film dynamics. A rotation of a ripple pattern by ${90}^{°}$ occurs in the system evolution when nonlinearities are strongly suppressed along one direction. This effect originates in nonlinear parameter renormalization at different rates in the two system dimensions, showing a dynamic interplay between scale invariance and wavelength selection. Potential experimental realizations of this phenomenon are identified.

We introduce a broad class of analytically solvable processes on networks. In the special case, they reduce to random walk and consensus process, the two most basic processes on networks. Our class differs from previous models of interactions (such as the stochastic Ising model, cellular automata, infinite particle systems, and the voter model) in several ways, the two most important being (i) the model is analytically solvable even when the dynamical equation for each node may be different and the network may have an arbitrary finite graph and influence structure and (ii) when local dynamics is described by the same evolution equation, the model is decomposable, with the equilibrium behavior of the system expressed as an explicit function of network topology and node dynamics.

In temporal networks, where nodes interact via sequences of temporary events, information or resources can only flow through paths that follow the time ordering of events. Such temporal paths play a crucial role in dynamic processes. However, since networks have so far been usually considered static or quasistatic, the properties of temporal paths are not yet well understood. Building on a definition and algorithmic implementation of the average temporal distance between nodes, we study temporal paths in empirical networks of human communication and air transport. Although temporal distances correlate with static graph distances, there is a large spread, and nodes that appear close from the static network view may be connected via slow paths or not at all. Differences between static and temporal properties are further highlighted in studies of the temporal closeness centrality. In addition, correlations and heterogeneities in the underlying event sequences affect temporal path lengths, increasing temporal distances in communication networks and decreasing them in the air transport network.

Interesting and exotic surface wave patterns have regularly been observed in the Faraday experiment. Although symmetry arguments provide a qualitative explanation for the selection of some of these patterns (e.g., superlattices), quantitative analysis is hindered by mathematical difficulties inherent in a time-dependent, free-boundary Navier-Stokes problem. More tractable low viscosity approximations are available, but these do not necessarily capture the moderate viscosity regime of the most interesting experiments. Here we focus on weakly nonlinear behavior and compare the scaling results derived from symmetry arguments in the low viscosity limit with the computed coefficients of appropriate amplitude equations using both the full Navier-Stokes equations and a reduced set of partial differential equations due to Zhang and Vinãls. We find the range of viscosities over which one can expect “low viscosity” theories to hold. We also find that there is an optimal viscosity range for locating superlattice patterns experimentally—large enough that the region of parameters giving stable patterns is not impracticably small, yet not so large that crucial resonance effects are washed out. These results help explain some of the discrepancies between theory and experiment.

We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible.

Traveling clusters are ubiquitously observed in the Hamiltonian mean-field model for a wide class of initial states, which are not predicted to become spatially inhomogeneous states by nonequilibrium statistical mechanics and by nonlinear Landau damping. To predict such a cluster state from a given initial state, we combine nonequilibrium statistical mechanics and a construction method of Bernstein-Greene-Kruskal (BGK) waves with the aid of phenomenological assumptions. The phenomenological theory is partially successful, and the theoretically constructed cluster states are in good agreement with $N$-body simulations. Robustness of the theory is also discussed for unsuccessful initial states.

We study the dynamics of a pair of parametrically driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by G. Kovačič and S. Wiggins [Physica D 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Šilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Šilnikov orbits are confirmed numerically.

A model is presented to explain the development of flow channels within the cytoplasm of the plasmodium of the giant amoeba Physarum polycephalum. The formation of channels is related to the development of a self-organizing tubular network in large cells. Experiments indicate that the flow of cytoplasm is involved in the development and organization of these networks, and the mathematical model proposed here is motivated by recent experiments involving the observation of development of flow channel in small cells. A model of pressure-driven flow through a polymer network is presented in which the rate of flow increases the rate of depolymerization. Numerical solutions and asymptotic analysis of the model in one spatial dimension show that under very general assumptions this model predicts the formation of channels in response to flow.

The static and dynamic (complex) shear viscosity of a single-layer dusty plasma is measured by applying, respectively, a stationary and a periodically modulated shear stress, induced by the light pressure of manipulating laser beams. Under static conditions we observe a decrease of the viscosity with increasing shear rate, the so-called shear-thinning behavior. Under oscillating shear both the magnitude and the ratio of the dissipative and elastic contributions to the complex viscosity show strong frequency dependence, as the system changes from viscous to elastic in nature with increasing excitation frequency. Accompanying molecular dynamics simulations explain and support the experimental observations.

Recent investigations of dust-density waves in a dusty plasma under microgravity conditions [K. O. Menzel et al., Phys. Rev. Lett. 104, 235002 (2010)] showed that the wave field consists of distinct regions of different frequencies. These so-called frequency clusters are known from simulations of chains of mutually coupled van der Pol oscillators. The behavior of distinct oscillators adjacent to the cluster boundaries were studied numerically. The interaction of these oscillators leads to periodic frequency pulling, a typical feature of driven van der Pol oscillators that is also observed in our experiments.

By means of a photoelastic method, we access the visualization of acoustic waves propagating in a one-dimensional array of noncohesive cylinders. As pointed by Nesterenko in the case of spherical grains [V. F. Nesterenko, J. Appl. Mech. Tech. Phys. 24, p. 567 (1983)], the nonlinearity of the contact law between the grains induces a dependence of the wave velocity both on its amplitude and on the confinement force. Our experimental method allows one to access the evolution in time of the internal state of stress of individual grains with excellent accuracy. We show that the velocity of the sound presents two regimes as a function of the confining force. For low forces, the dependence is strongly nonlinear, while it weakens for higher forces. By means of the direct visualization of the contact zone, we show that both micro- and macroscale imperfections of the surface of contact explain the low forces behavior. We test the consistency of our experimental findings results with both the theoretical expectations and with the experimental determination of the force-displacement dependence. We show, moreover, that the main damping process originates in solid friction.

A method for adiabatic control of envelope solitons in the driven nonlinear Schrödinger equation is developed. The approach is based on the autoresonant effect, when the soliton is captured (“phase locked”) by a two-phase resonant driving with slowly varying frequencies. Threshold conditions for amplitudes and variation rates of the driving required for the control of both the amplitude and the velocity of the soliton are found. Numerical simulations demonstrate that the method allows one to control solitons for a long time according to a given scenario, while the threshold conditions are fulfilled locally.

We study numerically rogue waves in dissipative systems, taking as an example a unidirectional fiber laser in a nonstationary regime of operation. The choice of specific set of parameters allows the laser to generate a chaotic sequence of pulses with a random distribution of peak amplitudes. The probability density function for the intensity maxima has an elevated tail at higher intensities. We have found that the probability of producing extreme pulses in this setup is higher than in any other system considered so far.

Based on Mie scattering theory and the assumption that the particle is electrically small in size ($k_{t}a\ll 1,k_{m}a\ll 1$), an analytic relationship between radial and tangential permittivity parameters has been established for achieving minimal scattering from an arbitrary rotationally uniaxial anisotropic spherical object incident by a plane wave. Analysis of fields in both the far- and near-field zones indicates that the derived relation is not only valid for electrically small particles, but also applicable to larger ones whose sizes are comparable with the free space wavelength after slight adjustments in parameters. Furthermore, it is observed that the dielectric spherical particle of reduced tangential permittivity yields better transparency performance than the design using a plasmonic cover by Alu and Engheta [Phys. Rev. E 72, 016623 (2005)]. As such, particles with carefully engineered radial anisotropy are transparent without using any coating and are ideal for applications with space constraint and stringent transparency criteria.

The polarization-induced charges of a dielectric sphere are studied for charged colloidal systems in electrolyte solutions with a primitive model. The method of constructing multiple-image charges is used to approximate the polarization potential of a microion outside the sphere; it is based on a numerical discretization of the potential’s analytical integral representation, and can systematically approximate the exact potential with desired accuracy by varyiation of the number of point images. Different aspects of the image effects are then investigated by Monte Carlo simulations for several colloidal systems, in both salt-free and salty environments. Furthermore, we studied the influence of discrete surface charges of different valences, and demonstrate that the polarization charges can significantly strengthen charge reversal for the colloid-microion complex, especially for multivalent interfacial ions.

The time-dependent probability distribution function of the height for the Kardar-Parisi-Zhang equation with sharp wedge initial conditions has been obtained recently as a convolution between the Gumbel distribution and a difference of two Fredholm determinants. We evaluate numerically this distribution over the whole time span. The crossover from the short time behavior, which is Gaussian, to the long time behavior, which is governed by the Gaussian unitary ensemble (GUE) Tracy-Widom distribution, is clearly visible.

In this work we address the problem of controlling a complex network toward a synchronous evolution by using the pinning-control strategy. In considering this control approach, we investigate the effect of the number of pinned nodes and how the strategy performance behaves in accordance with the type of nodes that are chosen to be controlled. The roles of the control and coupling gains are also discussed.

We measure the anisotropy of the crystal-fluid interfacial free energy in soft colloidal systems. A temperature gradient is used to direct crystal nucleation and control the growth of large single crystals in order to achieve well-equilibrated crystal-fluid interfaces. Confocal microscopy is used to follow both the growth process and the equilibrium crystal-fluid interface at the particle scale: heterogeneous crystal nucleation, the advancing interface, and the stationary equilibrium interface. We use the measured growth velocity to determine the chemical potential difference between crystal and fluid phases. Well-equilibrated, large crystal-fluid interfaces are then used to determine the interfacial free energy and its anisotropy directly from thermally excited interface fluctuations. We find that while the measured average interfacial free energy is in good agreement with values found in simulations, the anisotropy is significantly larger than simulation values. Finally, we investigate the effect of impurities on the advancing interface. We determine the critical force needed to overcome impurity particles from the local interface curvature.

We propose a model of mass-conserving heterogeneous nucleation to describe the dynamics of ligand-receptor binding in closed cellular compartments. When the ligand dissociation rate is small, competition among receptors for free ligands gives rise to two very different long-time ligand-receptor cluster-size distributions. Cluster sizes first plateau to a long-lived, initial-condition-dependent, “metastable” distribution, and coarsen only much later to a qualitatively different equilibrium one. Surprisingly, we also find parameters for which a very special subset of clusters have equal metastable and equilibrium sizes, appearing to equilibrate much faster than the rest. Our results provide a quantitative framework for ligand-binding kinetics and suggest a mechanism by which different clusters can approach their equilibrium sizes in unexpected ways.