We consider the canonical ensemble of multilayered constrained Erdos-Renyi networks (CERN) and regular random graphs (RRG), where each layer represents graph vertices painted in a specific color. We study the critical behavior in such networks under changing the fugacity, µ, which controls the number of monochromatic triads of nodes. The behavior of considered systems is investigated via the spectral properties of the adjacency and Laplacian matrices of corresponding networks. For some wide region of µ we find the formation of a finite plateau in the number of the intercolor links, which exactly matches the finite plateau for the algebraic connectivity of the network (the value of the first non-vanishing eigenvalue of the Laplacian matrix, λ2). We claim that at the plateau the restoring of the spontaneously broken Z2 symmetry by the mechanism of modes collectivization in clusters of different colors occurs. The phenomena of a finite plateau formation holds for the polychromatic (multilayer) networks with M > 2 colors.

We study theoretically and by means of molecular dynamics (MD) simulations the generation of mechanical force by grafted polyelectrolytes in an external electric field, which favors its adsorption on the grafting plane. The force arises in deformable bodies linked to the free end of the chain. Varying the field, one controls the length of the non-adsorbed part of the chain and hence the deformation of the target body, i.e., the arising force too. We consider target bodies with a linear force-deformation relation and with a Hertzian one. While the first relation models a coiled Gaussian chain, the second one describes the force response of a squeezed colloidal particle. The theoretical dependencies of generated force and compression of the target body on applied field agree very well with the results of MD simulations. The analyzed phenomenon may play an important role in a future nano-machinery, e.g. it may be used to design nano-vices to fix nano-sized objects.

The dynamics of coherent nonlinear wave groups is shown to be drastically different from the classical scenario of weakly nonlinear wave interactions. The coherent groups generate nonresonant (bound) waves which can be synchronized with other linear waves. By virtue of the revealed mechanism, the groups may emit waves with similar or different lengths, which propagate in the same or opposite direction.

We analyzed a generic relaxation oscillator under moderately strong forcing at a frequency much greater that the natural intrinsic frequency of the oscillator. Additionally, the forcing is of the same sign and, thus, has a nonzero average, matching neuroscience applications. We found that, first, the transition to high-frequency synchronous oscillations occurs mostly through periodic solutions with virtually no chaotic regimes present. Second, the amplitude of the high-frequency oscillations is large, suggesting an important role for these oscillations in applications. Third, the 1:1 synchronized solution may lose stability, and, contrary to other cases, this occurs at smaller, but not at higher frequency differences between intrinsic and forcing oscillations. We analytically built a map that gives an explanation of these properties. Thus, we found a way to substantially “overclock” the oscillator with only a moderately strong external force. Interestingly, in application to neuroscience, both excitatory and inhibitory inputs can force the high-frequency oscillations.

We describe the method for finding the non-Gaussian tails of the probability distribution function (PDF) for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration—the *instanton*. As an example, we examine the correlation functions of the passive scalar *u* advected by a large-scale velocity field δ correlated in time. We find the instanton determining the tails of the generating functional, and show that it is different from the instanton that determines the probability distribution function of high powers of *u*. We discuss the simplest instantons for the Navier-Stokes equation. © *1996 The American Physical Society.*

A system of equations describing motion of dust particles in gas discharge plasma is formulated. This system

is developed for a monolayer of dust particles with an account of dust particle charge fluctuations and features

of the discharge near-electrode layer. Molecular dynamics simulation of the dust particles system is performed.

A mechanism of dust particle average kinetic energy increase is suggested on the basis of theoretical analysis of

the simulation results. It is shown that heating of dust particles’ vertical motion is initiated by forced oscillations

caused by the dust particles’ charge fluctuations. The process of energy transfer from vertical to horizontal motion

is based on the phenomenon of the parametric resonance. The combination of parametric and forced resonances

explains the abnormally high values of the dust particles’ kinetic energy. Estimates of frequency, amplitude, and

kinetic energy of dust particles are close to the experimental values.

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in *E. coli*. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

We present a proof of the monotonic entropy growth for a nonlinear discrete-time model of a random market. This model, based on binary collisions, also may be viewed as a particular case of Ulam’s redistribution of energy problem. We represent each step of this dynamics as a combination of two processes. The first one is a linear energy-conserving evolution of the two-particle distribution, for which the entropy growth can be easily verified. The original nonlinear process is actually a result of a specific “coarse graining” of this linear evolution, when after the collision one variable is integrated away. This coarse graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth. The combination of these two factors produces the required result which is obtained only by means of information theory inequalities.

The dynamics of domain walls in optical bistable systems with pump and loss is considered. It is shown that an oscillating component of the pump affects the average drift velocity of the domain walls. The cases of harmonic and biharmonic pumps are considered. It is demonstrated that in the case of biharmonic pulse the velocity of the domain wall can be controlled by the mutual phase of the harmonics. The analogy between this phenomenon and the ratchet effect is drawn. Synchronization of the moving domain walls by the oscillating pump in discrete systems is studied and discussed.

The appearance of vortex filaments, the power-law dependence of velocity and vorticity correlations and their multiscaling behavior are derived from the Navier-Stokes equation. This is possible due to interpretation of the Navier-Stokes equation as an equation with multiplicative noise and remarkable properties of random matrix products.

The influence of periodic shear deformation on nonaffine atomic displacements in an amorphous solid is examined via molecular dynamics simulations. We study the three-dimensional Kob-Andersen binary mixture model at a finite temperature. It is found that when thematerial is periodically strained, most of the atoms undergo repetitive nonaffine displacements with amplitudes that are broadly distributed. We show that particles with large amplitudes of nonaffine displacements are organized into compact clusters. With increasing strain amplitude, spatial correlations of nonaffine displacements become increasingly long-ranged, although they remain present even in a quiescent system due to thermal fluctuations.

The present work follows our previous study dealing with a new type of synchronization in a system of two weakly coupled generalized van der Pol–Duffing autogenerators. The essence of the effect revealed is that the synchronized oscillations are not stationary but accompanied by the most intensive energy exchange between the oscillators. The phase shift between the generators remains constant most of the time, except for vanishingly small transitional intervals. The current analysis deals with a generalized model in order to clarify the frequency detuning effect. We found that varying the frequency detuning, nonlinearity, and dissipation parameters can lead to structural changes in phase diagrams of the energy exchange dynamics, with important transitions from the intensive energy exchange to its localization on one of the two oscillators. The main conclusion is that stationary and nonstationary synchronizations associate with nonlinear normal and local modes, respectively. The analysis uses phase plane diagrams, including the concept of limiting phase trajectories, whose role in nonstationary synchronization appears to be similar to the role of nonlinear normal modes in conventional stationary states.

Long-scale dynamic fluctuation phenomena in freely suspended films is analyzed. We consider isotropic films that, say, can be pulled from bulk smectic-

A liquid crystals. The key feature of such objects is possibility of bending deformations of the film. The bending (also known as flexular) mode turns out to be anomalously weakly attenuated. In the harmonic approximation there is no viscous-like damping of the bending mode, proportional to q 2 ( q is the wave vector of the mode), since it is forbidden by the rotational symmetry. Therefore, the bending mode is strongly affected by nonlinear dynamic fluctuation effects. We calculate the dominant fluctuation contributions to the damping of the bending mode due to its coupling to the inplane viscous mode, which restores the viscous-like q 2 damping of the bending mode. Our calculations are performed in the framework of the perturbation theory where the coupling of the modes is assumed to be small, then the bending mode damping is relatively weak. We discuss our results in the context of existing experiments and numeric simulations of the freely suspended films and propose possible experimental observations of our predictions.We investigate an intermittent stochastic process in which diffusive motion with a time-dependent diffusion coefficient, D(t ) ∼ t α−1, α > 0 (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either an exponential or a power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a nonrenewal one.We discuss the mean squared displacement of a particle and the probability density function of its positions in this process.We show that scaled Brownian motion with resetting demonstrates rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle’s displacement infree motion. The motion of particles can remain almost unaffected by resetting but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. This behavior is compared to the situation [discussed in the companion paper; A. S. Bodrova et al., Phys. Rev. E 100, 012120 (2019)] in which the memory of the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different.

The low- and high-amplitude oscillations in the system of three nonlinear coupled pendula (trimer) are analyzed beyond the quasilinear approximation. The considered oscillations are fundamental for many models of the energy exchange processes in physical, mechanical, and biological systems, in particular, for the torsional vibrations of flexible polymers or DNA's double strands. We obtained the conditions of the basic stationary solutions' stability. These solutions correspond to the nonlinear normal modes (NNMs), the instability of which leads to the appearance of localized NNMs (stationary energy localization). Using an asymptotic procedure, we reduce the dimension of the system's phase space that allows us to analyze the energy exchange between pendula in the slow timescale and to reveal periodic interparticle energy exchange and nonstationary energy localization. It has been shown recently that essentially nonstationary resonance processes of this type are adequately described in terms of the limiting phase trajectories (LPTs) corresponding to beatings between the oscillators or coherence domains in the slow timescale. Moreover, it turns out that criteria of the transition to the stationary and nonstationary energy localization can be formulated as the bifurcation conditions for NNMs and LPTs, respectively. The trimer under consideration is a nonintegrable system, and therefore its equations of motion is only after dimensions reduction can be analyzed by the Poincare sections method. Finally, we aim to study the highly nonstationary regimes, which correspond to beatinglike periodic or quasiperiodic recurrent energy exchange between the pendula.

We compute analytically the mean number of common sites, WN(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N−d) plane, there are three distinct regimes for the asymptotic large-t growth of WN(t). These three regimes are separated by two critical lines d=2 and d=dc(N)=2N/(N−1) in the (N-d) plane. For d<2, WN(t)∼td/2 for large t (the N dependence is only in the prefactor). For 2<d<dc(N), WN(t)∼tν where the exponent ν=N−d(N−1)/2 varies with N and d. Ford>dc(N), WN(t)→const as t→∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get WN(t)∼t/[lnt]N, while for d=dc(N), WN(t)∼lnt for large t. Our analytical predictions are verified in numerical simulations.

We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrodinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed by the NLS.