We study collective synchronization in a large number of coupled oscillators on various complex networks. In particular, we focus on the relaxation dynamics of the synchronization, which is important from the viewpoint of information transfer or the dynamics of system recovery from a perturbation. We measure the relaxation time $\tau$ that is required to establish global synchronization by varying the structural properties of the networks. It is found that the relaxation time in a strong-coupling regime $\left(K>K_c\right)$ logarithmically increases with network size $N$, which is attributed to the initial random phase fluctuation given by $\mathcal{O}\left(N^{-1/2}\right)$. After elimination of the initial-phase fluctuation, the relaxation time is found to be independent of the system size; this implies that the local interaction that depends on the structural connectivity is irrelevant in the relaxation dynamics of the synchronization in the strong-coupling regime. The relaxation dynamics is analytically derived in a form independent of the system size, and it exhibits good consistency with numerical simulations. As an application, we also explore the recovery dynamics of the oscillators when perturbations enter the system.

We study exact four-wave resonances among gravity water waves in a square box with periodic boundary conditions. We show that these resonant quartets are linked with each other by shared Fourier modes in such a way that they form independent clusters. These clusters can be formed by two types of quartets: (1) Angle resonances which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) scale resonances which are much more rare but which are the only ones that can transfer energy between different scales. We find such resonant quartets and their clusters numerically on the set of $1000\times 1000$ modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wave-field evolution. Finite box effects and associated resonant interaction among discrete wave modes appear to be important in most numerical and laboratory experiments on the deep water gravity waves, and our work is aimed at aiding the interpretation of the experimental and numerical data.

We study the role of the quench temperature $T_f$ in the phase-ordering kinetics of the Ising model with single spin flip in $d=2,3$. Equilibrium interfaces are flat at $T_f=0$, whereas at $T_f>0$ they are curved and rough (above the roughening temperature in $d=3$). We show, by means of scaling arguments and numerical simulations, that this geometrical difference is important for the phase-ordering kinetics as well. In particular, while the growth exponent $z=2$ of the size of domains $L\left(t\right)\sim t^{1/z}$ is unaffected by $T_f$, other exponents related to the interface geometry take different values at $T_f=0$ or $T_f>0$. For $T_f>0$ a crossover phenomenon is observed from an early stage where interfaces are still flat and the system behaves as at $T_f=0$, to the asymptotic regime with curved interfaces characteristic of $T_f>0$. Furthermore, it is shown that the roughening length, although subdominant with respect to $L\left(t\right)$, produces appreciable correction to scaling up to very long times in $d=2$.

We consider the two-dimensional randomly site diluted Ising model and the random-bond $\pm J$ Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.

Numerous experimental and computational studies have been carried out in recent years to understand the mechanisms governing the compaction of granular systems. Here the problem is further investigated from a different perspective. We compact spheres by a vibrational annealing method and show how the interactions between them and the walls determine the final structure. Dry spheres self-assemble only in body-centered-tetragonal structures, while cohesive ones surpass such density and reach the most compact face-centered-cubic phase. We argue that such polymorphism is due to a molecularlike behavior induced by a compensation mechanism between free and vibrational energies.

We investigate the evolution of cooperation on a nongrowth dynamic network model with a death-birth dynamics based on tournament selection. In the limit of large population size, the equilibrium cooperator density is well described by our mean field approximations and inversely related to the average degree in the system. Small populations are also examined and found to deviate considerably from their expected mean field behavior. An expanded replicator equation incorporating Gaussian fluctuations in the strategy densities is then constructed, with its output agreeing well with our simulation data for all sizes. We also briefly comment on the role of strategy mutation in sustaining polymorphic populations in small systems.

Birdsong is a model system for learned vocal behavior with remarkable parallels to human vocal development and sound production mechanisms. Upper vocal tract filtering plays an important role in human speech, and its importance has recently also been recognized in birdsong. However, the mechanisms of how the avian sound source might contribute to spectral richness are largely unknown. Here we show in the most widely studied songbird, the zebra finch (Taeniopygia guttata), that the broad range of upper harmonic content in different low-frequency song elements is the fingerprint of the dynamics displayed by its vocal apparatus, which can be captured by a two-dimensional dynamical model. As in human speech and singing, the varying harmonic content of birdsong is not only the result of vocal tract filtering but of a varying degree of tonality emerging from the sound source. The spectral content carries a strong signature of the intrinsic dynamics of the sound source.

The results of numerical simulations indicate that deterministic walks with inverse-square power-law scaling are a robust emergent property of predators that use chemotaxis to locate randomly and sparsely distributed stationary prey items. It is suggested that chemotactic destructive foraging accounts for the apparent Lévy flight movement patterns of Oxyrrhis marina microzooplankton in still water containing prey items. This challenges the view that these organisms are executing an innate optimal Lévy flight searching strategy. Crucial for the emergence of inverse-square power-law scaling is the tendency of chemotaxis to occasionally cause predators to miss the nearest prey item, an occurrence which would not arise if prey were located through the employment of a reliable cognitive map or if prey location were visually cued and perfect.

We demonstrate that rare intense events in the complex Ginzburg-Landau equation can be suppressed by using only observation-based control. Our control strategy eliminates the requirements of a precise system model and prediction. Analytic insights for guiding the control and numerical verification are obtained. The issue of time delay is also addressed. We expect our results to provide insight into the challenging problem of harnessing spatially extended complex systems.

We investigate the coevolutionary dynamics of opinions and networks based upon majority-preference (MP) and minority-avoidance (MA) rules. Under MP, individuals adopt the majority opinion among their neighbors; while in MA individuals can break the link to one holding a minority and different opinion, and rewire either to neighbors of their neighbors with the same opinion or to a random one from the whole population except their nearest neighbors. We study opinion formation as a result of combination of these two competing rules, with a parameter tuning the balance between them. We find that the underlying network can be self-organized into connected communities with like-minded individuals belonging to the same group; thus a broad variety of opinions coexist. Diverse opinions disappear in a population in which all individuals share a uniform opinion, when the model parameter exceeds a critical value. Furthermore, we show that an increasing tendency to redirect to neighbors of neighbors is more likely to result in a consensus of opinion.

Bunches of streamers form the early stages of sparks and lightning but theory presently concentrates on single streamers or on coarse approximations of whole breakdown trees. Here a periodic array of interacting streamer discharges in a strong homogeneous electric field is studied in density or fluid approximation in two dimensions. If the period of the streamer array is small enough, the streamers do not branch, but approach uniform translation. When the streamers are close to the branching regime, the enhanced field at the tip of the streamer is close to $2E_{\infty }$, where $E_{\infty }$ is the homogeneous field applied between the electrodes. We discuss a moving boundary approximation to the density model. This moving boundary model turns out to be essentially the same as the one for two-fluid Hele-Shaw flows. In two dimensions, this model possesses a known analytical solution. The shape of the two-dimensional interacting streamers in uniform motion obtained from the PDE simulations is actually well fitted by the analytically known “selected Saffman-Taylor finger.” This finding helps to understand streamer interactions and raises new questions on the general theory of finger selection in moving boundary problems.

We report on bistable mode-locking in a semiconductor disk laser. The disk laser mode-locked with a semiconductor saturable absorber is investigated for different designs of the gain medium that allow the hysteresis loop to be controlled. Hysteresis formation in the pulsed regime of a semiconductor oscillator with saturable absorption and unsaturated gain is discussed qualitatively. The laser represents an attractive setup for generation and manipulation of dissipative solitons and observation of their interaction.

We present a classical approach to a mixture of compressible fluids when each constituent has its own temperature. The introduction of an average temperature together with the entropy principle dictates the classical Fick law for diffusion and also novel constitutive equations associated with the difference of temperatures between the components. The constitutive equations fit with results recently obtained through a Maxwellian iteration procedure in extended thermodynamics theory of multitemperature mixtures. The differences of temperatures between the constituents imply the existence of a dynamical pressure even if the fluids have a zero bulk viscosity. The nonequilibrium dynamical pressure can be measured and may be convenient in several physical situations, such as, for example, in cosmological circumstances where—as many authors assert—a dynamical pressure played a major role in the evolution of the early universe.

The anisotropy properties of magnetohydrodynamic turbulence in a sheared magnetic field are analyzed through a three-dimensional numerical simulation that reproduces the linear and nonlinear stage of a tearing instability. Far from the current sheet, the energy spectrum develops perpendicularly to the local magnetic field, as in homogeneous configurations. Within the current sheet, the spectrum anisotropy is also affected by the structure of unstable modes. With increasing time, the configuration becomes more turbulent, the former effect disappears, and the energy cascade takes place perpendicularly to the local magnetic field. The local spectrum becomes increasingly anisotropic while the spatially integrated spectrum tends to isotropize. There is the possibility that these properties could be used to identify the nonlinear stage of magnetic reconnection in space and laboratory plasmas, as well as to identify the particle transport regime in the considered magnetic configuration.

We show that when an individual particle is dragged through an assembly of other particles in the presence of quenched disorder, a viscous decoupling transition occurs between the dragged particle and the surrounding particles which is controlled by the quenched disorder. A counterintuitive consequence of this transition is that the velocity of the dragged particle can be increased by increasing the strength or density of the quenched disorder. The decoupling transition can also occur when the external drive on the dragged particle is increased and is observable as a clear signature in the velocity-force response.

We report that a polymer molecule can be trapped spatially and conformationally using a microflow that has at least two stagnation points (or two points with equal velocity) and a net flow orthogonal to the line connecting them. Examples include a Taylor vortex flow and an electro-osmotic flow in a channel with surfaces that have a sinusoidal charge. Simulating the motion of a polymer molecule in these flows using Brownian dynamics, we find that such flows produce a curved polymer conformation, leading to an elastic force that drives migration against the flow, thus stabilizing this conformation. Simulations with hydrodynamic interactions confirm these predictions and show that there exists a repulsive interaction between two trapped polymers.

In this paper, the acoustical memory effect is studied from 2.5 to 10 MHz. This acoustical memory effect has been observed in both columnar and cubic samples. The observed phenomenon of amplitude hysteresis of the acoustical memory signal suggests nonclassical nonlinear properties, caused by the structural inhomogeneity and inevitable flaws in ${\text{LiNbO}}_3$. Experimental results also show that the amplitude of the acoustical memory signal is enhanced by increasing the temperature at these frequencies. The shape of the memory signal changes greatly if the driving voltage is sufficiently high or each burst contains enough cycles; second and even third parts of the acoustical memory signals may appear. Amplitude hysteresis of the fundamental and second harmonic of the direct wave can also be found. Some experimental results are qualitatively explained in this paper.

Cardiac alternans, a beat-to-beat alternation in action potential duration (at the cellular level) or in electrocardiogram morphology (at the whole heart level), is a marker of ventricular fibrillation, a fatal heart rhythm that kills hundreds of thousands of people in the United States each year. Investigating cardiac alternans may lead to a better understanding of the mechanisms of cardiac arrhythmias and eventually better algorithms for the prediction and prevention of such dreadful diseases. In paced cardiac tissue, alternans develops under increasingly shorter pacing period. Existing experimental and theoretical studies adopt the assumption that alternans in homogeneous cardiac tissue is exclusively determined by the pacing period. In contrast, we find that, when calcium-driven alternans develops in cardiac fibers, it may take different spatiotemporal patterns depending on the pacing history. Because there coexist multiple alternans solutions for a given pacing period, the alternans pattern on a fiber becomes unpredictable. Using numerical simulation and theoretical analysis, we show that the coexistence of multiple alternans patterns is induced by the interaction between electrotonic coupling and an instability in calcium cycling.

We consider a pair of random heteropolymer chains with quenched primary sequences. For this system we have analyzed the dependence of average ground state energy per monomer $\mathcal{E}$ on chain length $n$ in the ensemble of chains with uniform distribution of primary sequences of monomers. Every monomer of the first (second) chain is randomly and independently chosen with the uniform probability distribution $p=1/c$ from a set of $c$ different types $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, $\mathit{D}$,... $\left(A^{\prime },B^{\prime },C^{\prime },D^{\prime },\dots \right)$. Monomers of the first chain could form saturating reversible bonds with monomers of the second chain. The bonds between similar monomer types (such as $A–A^{\prime }$, $B–B^{\prime }$, $C–C^{\prime }$, etc.) have the attraction energy $u$, while the bonds between different monomer types (such as $A–B^{\prime }$, $A–D^{\prime }$, $B–D^{\prime }$, etc.) have the attraction energy $v$. The main attention is paid to the computation of the normalized free energy $\mathcal{E}\left(n\right)$ for intermediate chain lengths $n$ and different ratios $a=v/u$ at sufficiently low temperatures, when the entropic contribution of the loop formation is negligible compared to direct energetic interactions between chain monomers, and when the partition function of the chains is dominated by the ground state. The performed analysis allows one to derive the force $f\left(x\right)$ which is necessary to apply for unzipping of two random heteropolymers of equal lengths whose ends are separated by the distance $x$, averaged over all equally distributed primary structures at low temperatures for fixed values $a$ and $c$.

We study experimentally the influence of confinement on the penetration depth of impacting spheres into a granular medium contained in a finite cylindrical vessel. The presence of close lateral walls reduces the penetration depth, and the characteristic distance for these lateral wall effects is found to be of the order of one sphere diameter. The influence of the bottom wall is found to have a much shorter range.