We address the problem of community detection in networks by introducing a general definition of Markov stability, based on the difference between the probability fluxes of a Markov chain on the network at different timescales. The specific implementation of the quality function and the resulting optimal community structure thus become dependent both on the type of Markov process and on the specific Markov times considered. For instance, if we use a natural Markov chain dynamics and discount its stationary distribution (that is, we take as reference process the dynamics at infinite time) we obtain the standard formulation of the Markov stability. Notably, the possibility to use finite-time transition probabilities to define the reference process naturally allows detecting communities at different resolutions, without the need to consider a continuous-time Markov chain in the small time limit. The main advantage of our general formulation of Markov stability based on dynamical flows is that we work with lumped Markov chains on network partitions, having the same stationary distribution of the original process. In this way the form of the quality function becomes invariant under partitioning, leading to a self-consistent definition of community structures at different aggregation scales.

As the constituent particles of a flock are polar and in a driven state, their interactions must, in general, be fore-aft asymmetric and nonreciprocal. Within a model that explicitly retains the classical spin angular momentum field of the particles we show that the resulting asymmetric contribution to interparticle torques, if large enough, leads to a buckling instability of the flock. More precisely, this asymmetry also yields a natural mechanism for a difference between the speed of advection of polarization information along the flock and the speed of the flock itself, concretely establishing that the absence of detailed balance, and not merely the breaking of Galilean invariance, is crucial for this distinction. To highlight this we construct a model of asymmetrically interacting spins fixed to lattice points and demonstrate that the speed of advection of polarization remains nonzero. We delineate the conditions on parameters and wave number for the existence of the buckling instability. Our theory should be consequential for interpreting the behavior of real animal groups as well as experimental studies of artificial flocks composed of polar motile rods on substrates.

We consider ultrashort light pulse propagation through an inhomogeneous monomodal optical fiber exhibiting higher-order dispersive effects. Wave propagation is governed by a generalized nonlinear Schrödinger equation with varying second-, third-, and fourth-order dispersions, cubic nonlinearity, and linear gain or loss. We construct a type of exact self-similar soliton solution that takes the structure of a dipole via a similarity transformation connected to the related constant-coefficients one. The conditions on the optical-fiber parameters for the existence of these self-similar structures are also given. The results show that the contribution of all orders of dispersion is an important feature to form this kind of self-similar dipole pulse shape. The dynamic behaviors of the self-similar dipole solitons in a periodic distributed amplification system are analyzed. The significance of the obtained self-similar pulses is also discussed. By performing numerical simulations, the self-similar soliton solutions are found to be stable under slight disturbance of the constraint conditions and the initial perturbation of white noise.

We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that this solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain an exact solution to the nonlocal NLSE using the same method and seed solution. The solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.

We analyze a model of two identical chemical oscillators coupled through diffusion of the slow variable. As a parameter is varied, a single oscillator undergoes a canard explosion—a transition from small amplitude, nearly harmonic oscillations to large-amplitude, relaxation oscillations over a very small parameter interval. In the coupled system, if the two oscillators have the same initial conditions, then the oscillators remain synchronized and exhibit the same canard behavior observed for the single oscillator. If the oscillators are separated initially, then in the region of the canard they display a variety of complex behaviors, including intermittent spiking, mixed-mode oscillation, and quasiperiodicity. Further variation of the parameter leads to a return to synchronized large-amplitude oscillation followed by a post-canard symmetry-breaking, in which one oscillator shows small-amplitude, complex behavior (mixed-mode oscillation, quasiperiodicity, chaos,...) while the other undergoes essentially periodic large amplitude behavior, resembling a master-slave scenario. We analyze the origins of this behavior by looking at several modified coupling schemes.

Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player–two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games—the Stag Hunt game, the Snowdrift game, the Prisoners' Dilemma game, and the Harmony game—we analytically and numerically analyze the delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcation bringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply the coexistence of the cooperators and the defectors; and in the games, where defection is a stable Nash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We find that while mutation alone can never lead to oscillatory cooperation state in two-player–two-strategy games, the delay can change the scenario. On the other hand, there are situations when delay alone cannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.

Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and three-dimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons, and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons, and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons. Moreover, we discuss how other material parameters can be used to determine soliton stability and how similar principles can be systematically applied to other liquid crystal solitons and solitons in other material systems.

Rayleigh-Taylor instability (RTI) under radiation background is commonly found in both engineering applications and natural phenomena. In the optically thin and incompressible limit, the corresponding problem can be simplified as an interface discontinuous acceleration (IDA) RTI problem, but to date has only been studied in the linear stage. In this paper, the entire IDA-RTI evolution was studied numerically and theoretically, particularly for the stages beyond the linear stage. The results show that the IDA-RTI problem is equivalent to the classical RTI with the effective acceleration $g_{\mathrm{eff}}^{*}$ that is introduced in this work. Moreover, our studies further show that IDA-RTI can occur if and only if $g_{\mathrm{eff}}^{*}>0$ (from heavy fluid to light fluid). This criterion means that IDA-RTI can occur when (i) heavy fluid supports (or accelerates) the light fluid or (ii) the two fluids have the same density, in contrast to the classical RTI problem. Moreover, the quasisteady bubble and spike velocities are theoretically predicted with quantitative accuracy, showing good agreement with the results of numerical simulations in a wide range of density ratios and acceleration configurations.

Assuming that we wish to measure the surface tension between two liquids by running a pendent drop experiment, we present calculations supporting the case for spinning the drop. For bridges, jets, etc., spinning a heavy fluid surrounded by a lighter fluid is strictly destabilizing. But we find that spinning a drop may be stabilizing and, if this is so, it leads to larger critical volumes, volumes where stability is lost, and thus more accurate measurements of surface tension. There are two observable patterns, one symmetric and the other unsymmetric, at the point of instability. The symmetric pattern leads to larger critical volumes. Our aim is to show how spinning can be used to achieve the symmetric pattern.

We carry out a numerical investigation of three-dimensional linear perturbations in a self-similar ablation wave in slab symmetry, representative of the shock transit phase of a pellet implosion in inertial confinement fusion (ICF). The physics of ablation is modeled by the equation of gas dynamics with a nonlinear heat conduction as an approximation for radiation transport. Linear perturbation responses of the flow, its external surface, and shock-wave front, when excited by external pressure or heat-flux perturbation pulses, are computed by fully taking into account the flow compressibility, nonuniformity, and unsteadiness. These responses show the effective propagation, at supersonic speeds, of perturbations from the flow external surface through the whole conduction region of the ablation wave, beyond its Chapman-Jouguet point, and up to the ablation front, after the birth of the ablation wave. This supersonic forward propagation of perturbations is evidenced by means of a set of appropriate pseudocharacteristic variables and is analyzed to be associated to the “heat-conductivity” waves previously identified by Clarisse et al. [J. Fluid Mech. 848, 219 (2018)]. Such heat-conductivity linear waves are found to prevail over heat diffusion as a feedthrough mechanism [Aglitskiy et al., Philos. Trans. R. Soc. A 368, 1739 (2010)] for perturbations of longitudinal characteristic lengths of the order of—or larger than—the conduction region size, and long transverse wavelengths with respect to this region size, and over time scales shorter to much shorter than the shock transit phase duration. This mechanism, which results from the dependency of the heat conductivity on temperature and density in conjunction with a flow temperature stratification, is expected to occur for other types of nonlinear heat conductions—e.g., electron heat conduction—as well as to be efficient at transmitting large scale perturbations from the surrounding of an ICF pellet to its inner compressed core at later times of its implosion. Besides, the proposed set of pseudocharacteristic variables are recommended for analyzing perturbation dynamics in an ablation flow as it furnishes additional propagation information over the fundamental linear modes of fluid dynamics [Kovásznay, J. Aeronautic. Sci. 20, 657 (1953)], especially in the flow conduction region.

Synchronization is a very generic phenomenon which can be encountered in a large variety of coupled dynamical systems. Being able to synchronize chaotic systems is strongly dependent on the nature of their coupling. Few attempts to explain such a dependency using observability and/or controllability were not fully satisfactory and synchronizability yet remained unexplained. Synchronizability can be defined as the range of coupling parameter values for which two nearly identical systems are fully synchronized. Our objective is here to investigate whether synchronizability can be related to the main rotation necessarily required for structuring any type of attractor, that is, whether synchronizability is significantly improved when the coupling variable is one of the variables involved in the main rotation. We thus propose a semianalytic procedure from a single isolated system to discard the worst variable for fully synchronizing two (nearly) identical copies of that system.

A density oscillator exhibits limit-cycle oscillations driven by the density difference of the two fluids. We performed two-dimensional hydrodynamic simulations with a simple model and reproduced the oscillatory flow observed in experiments. As the density difference is increased as a bifurcation parameter, a damped oscillation changes to a limit-cycle oscillation through a supercritical Hopf bifurcation. We estimated the critical density difference at the bifurcation point and confirmed that the period of the oscillation remains finite even around the bifurcation point.

We aim at an understanding of the dynamical properties of a periodically driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM) at zero temperature, which is capable of showing complex hysteresis. The system is a combination of a continuous (harmonic oscillator) and a discrete (RFIM) subsystem, which classifies it as a hybrid system. In this paper we focus on the hybrid nature of the system and consider only independent spins in quenched random local fields, which can already lead to complex dynamics such as chaos and multistability. We study the dynamic behavior of this system by using the theory of piecewise-smooth dynamical systems and discontinuity mappings. Specifically, we present bifurcation diagrams and Lyapunov exponents as well as results for the shape and the dimensions of the attractors and the self-averaging behavior of the attractor dimensions and the magnetization. Furthermore we investigate the dynamical behavior of the system for an increasing number of spins and the transition to the thermodynamic limit, where the system behaves like a driven harmonic oscillator with an additional nonlinear smooth external force.

We propose a method for the reconstruction of time-delayed feedback systems having unobserved variables from scalar time series. The method is based on the modified initial condition approach, which allows one to significantly reduce the number of starting guesses for an unobserved variable with a time delay. The proposed method is applied to the reconstruction of the Lang-Kobayashi equations, which describe the dynamics of a single-mode semiconductor laser with external optical feedback. We consider the case where only the time series of laser intensity is observable and the other two variables of the model are hidden. The dependence of the quality of the system reconstruction on the accuracy of assignment of starting guesses for unobserved variables and unknown laser parameters is studied. The method could be used for testing the security of information transmission in laser-based chaotic communication systems.

Previous studies claim that the dynamic behaviors of spiral waves are uniquely determined by the nature of the medium, which can be determined by control parameters. In this article, the authors break from the previous view and present an alternate stable state of spiral waves, named the excited state. The authors find that two states of the spiral wave switch to each other after a one-off pulse is applied to the medium. The dynamic behaviors of the two states are quite different, specifically, the spiral tip trajectory of the original spiral, which is named the ground-state spiral as observed in the previous studies, is a point, while the spiral tip trajectory of the excited-state spiral is a circle. Moreover, the authors study the trajectories of the spiral tip of spiral waves in both states after the pulse is applied and find two types of trajectories, a spiral trajectory and a spiral-inward-petal trajectory. The frequency of the spiral wave in the excited state is less than that in the ground state. The findings enrich the dynamics of pattern formation.

Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy, and engineering. Tipping points are critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change of the system. Many systems with tipping points can be modeled as networks of coupled multistable subsystems, e.g., coupled patches of vegetation, connected lakes, interacting climate tipping elements, and multiscale infrastructure systems. In such networks, tipping events in one subsystem are able to induce tipping cascades via domino effects. Here, we investigate the effects of network topology on the occurrence of such cascades. Numerical cascade simulations with a conceptual dynamical model for tipping points are conducted on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert networks. Additionally, we generate more realistic networks using data from moisture-recycling simulations of the Amazon rainforest and compare the results to those obtained for the model networks. We furthermore use a directed configuration model and a stochastic block model which preserve certain topological properties of the Amazon network to understand which of these properties are responsible for its increased vulnerability. We find that clustering and spatial organization increase the vulnerability of networks and can lead to tipping of the whole network. These results could be useful to evaluate which systems are vulnerable or robust due to their network topology and might help us to design or manage systems accordingly.

Designing the director alignment of a liquid crystal elastomer (LCE) is a key tuning approach for LCE based smart devices. In this paper, the spontaneous strain of the LCE with arbitrary three-dimensional director orientation is derived, and the governing equation for a simply supported LCE rectangular plate is established. By using the finite difference method, the bending configurations are obtained. Different from the freestanding case, three bending modes, that is, unimodal, bimodal, and trimodal modes, are observed for the simply supported rectangular LCE plate. The relation between the bending modes and the director orientation is established. This paper enhances the understanding and facilitates the design of LCE based intelligent light driven devices.