Standing Wave State Research Articles

Diverse phase transitions in Kuramoto model with adaptive mean-field coupling breaking the rotational symmetry

We consider the paradigmatic Kuramoto oscillators coupled via an adaptive mean-field variable that breaks the rotational symmetry. The evolution equation for the mean-field variable is governed by the linear ordinary differential equation corresponding to a low-pass filter with a time-scale parameter, which attenuates the mean-field signal by filtering its high frequency components. We find distinct phase transitions among the incoherent state, standing wave state and synchronized stationary state in both forward and reverse traces in the entire range of the time-scale parameter. In particular, one can observe two distinct continuous transitions, a continuous transition followed by a discontinuous transition, or a single discontinuous transition among the distinct dynamical states in the forward trace depending on the range of the time-scale parameter, which determines the degree of attenuation. Further, one can also observe two distinct continuous transitions or a single discontinuous desynchronization transition in the reverse trace depending on the time-scale parameter. We derive the evolution equations for the macroscopic order parameters governing the dynamics of the discrete set of coupled phase oscillators. We also deduce the analytical conditions for the stability of the Hopf, pitchfork and saddle–node bifurcations, and find them to be in consistent with the dynamical transitions observed using the simulation.

Synchronization in coupled oscillators with multiplex interactions

The study of synchronizations in coupled oscillators is very important for understanding the occurrence of self-organized behaviors in complex systems. In the traditional Kuramoto model that has been extensively applied to the study of synchronous dynamics of coupled oscillators, the interaction function among oscillators is pairwise. The multiplex interaction mechanism that describes triple or multiple coupling functions has been a research focus in recent years. When the multiplex coupling dominates the interactions among oscillators, the phase oscillator systems can exhibit the typical abrupt desynchronization transitions. In this paper, we extensively investigate the synchronous dynamics of the Kuramoto model with mean-field triple couplings. We find that the abrupt desynchronization transition is irreversible, i.e. the system may experience a discontinuous transition from coherent state to incoherent state as the coupling strength deceases adiabatically, while the reversed transition cannot occur by adiabatically increasing the coupling. Moreover, the coherent state strongly depends on initial conditions. The dynamical mechanism of this irreversibility is theoretically studied by using the self-consistency approach. The neutral stability of ordered state is also explained through analyzing the linear-stability of the incoherent state. Further studies indicate that the system may experience a cascade of desynchronized standing-wave transitions when the width of the distribution function of natural frequencies of oscillators is changed. At the critical coupling, the motion of coupled oscillators in high-dimensional phase space becomes unstable through the saddle-node bifurcation and collapses into a stable low-dimensional invariant torus, which corresponds to the standing-wave state. The above conclusions and analyses are further extended to the case of multi-peak natural-frequency distributions. The results in this work reveal various collective synchronous states and the mechanism of the transitions among these macroscopic states brought by multiplex coupling. This also conduces to the in-depth understanding of transitions among collective states in other complex systems.

Self-interaction of ultrashort pulses in an epsilon-near-zero nonlinear material at the telecom wavelength.

Dynamics of femtosecond pulses with the telecom carrier wavelength is investigated numerically in a subwavelength layer of an indium tin oxide (ITO) epsilon-near-zero (ENZ) material with high dispersion and high nonlinearity. Due to the subwavelength thickness of the ITO ENZ material, and the fact that the pulse's propagation time is shorter than its temporal width, multiple reflections give rise to self-interaction in both spectral and temporal domains, especially at wavelengths longer than at the ENZ point, at which the reflections are significantly stronger. A larger absolute value of the pulse's chirp strongly affects the self-interaction by redistributing energy between wavelengths, while the sign of the chirp affects the interaction in the temporal domain. It is also found that, when two identical pulses are launched simultaneously from both ends, a subwavelength counterpart of a standing-wave state can be established. It shows robust energy localization in the middle of the sample, in terms of both the spectral and temporal intensity distributions.

Nontrivial standing wave state in frequency-weighted Kuramoto model

Synchronization in a frequency-weighted Kuramoto model with a uniform frequency distribution is studied. We plot the bifurcation diagram and identify the asymptotic coherent states. Numerical simulations show that the system undergoes two first-order transitions in both the forward and backward directions. Apart from the trivial phase-locked state, a novel nonstationary coherent state, i.e., a nontrivial standing wave state is observed and characterized. In this state, oscillators inside the coherent clusters are not frequency-locked as they would be in the usual standing wave state. Instead, their average frequencies are locked to a constant. The critical coupling strength from the incoherent state to the nontrivial standing wave state can be obtained by performing linear stability analysis. The theoretical results are supported by the numerical simulations.

Synchronization of phase oscillators with frequency-weighted coupling.

Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

Electronic state of zigzag graphene nanoribbons

Based on the tight-binding model, the electronic state and band of zigzag graphene nanoribbons are given analytically by a new method. The results show that there are only two kinds electronic states, i.e., the standing wave state and edge state. For the standing wave state, the wave function is sine function and the vector is real; for the edge state, the wave function is hyperbolic sine function and the vector is complex, whose real part is 0 or π/2. The energy band is composed of the energy of standing wave state and the energy of edge state. The accurate ranges of infinite direction wave vector and energy of the edge state are deduced. Then we discuss the transition point between the edge state and the standing wave state and find that the two kinds of electronic states tend to the linear relationship regarding the site of carbon lattice in different ways at the phase transition point. When the width of two restricted boundary goes to infinity, the result of the limited graphene tends to the infinite case.

Electronic state of the limited graphene

The limited graphene means that two directions of graphene are limited, one is zigzag type boundary and the other is armchair type boundary. Based on the tight-binding model, the electronic state and band of the limited graphene are given analytically. The results show that there are only two kinds of electronic states, i.e., the standing wave state and edge state. For the standing wave state, the wave function is in the form of sine function in two directions; for the edge state, the wave function is in the form of hyperbolic sine function in the direction of armchair boundary and in the form of sine function in the direction of zigzag boundary. The band is composited of total carbon atom number N discrete eigenvalues. The expression of quantitativly calculating the number of eigenvalues of edge state is deduced. Through the density of states of the limited graphene we analyze the existence of the edge state and the consistency in the infinity case. The results from the analitical method are the same as the numerical resullts. When the width of two restricted boundary goes into infinity, the result of the limited graphene tends to that in the infinity case.

Nanographene and Graphene Edges: Electronic Structure and Nanofabrication

Graphene can be referred to as an infinite polycyclic aromatic hydrocarbon (PAH) consisting of an infinite number of benzene rings fused together. However, at the nanoscale, nanographene's properties lie in between those of bulk graphene and large PAH molecules, and its electronic properties depend on the influence of the edges, which disrupt the infinite π-electron system. The resulting modulation of the electronic states depends on whether the nanographene edge is the armchair or zigzag type, corresponding to the two fundamental crystal axes. In this Account, we report the results of fabricating both types of edges in the nanographene system and characterizing their electronic properties using a scanning probe microscope. We first introduce the theoretical background to understand the two types of finite size effects on the electronic states of nanographene (i) the standing wave state and (ii) the edge state which correspond to the armchair and zigzag edges, respectively. Most importantly, characterizing the standing wave and edge states could play a crucial role in understanding the chemical reactivity, thermodynamic stability and magnetism of nanosized graphene--important knowledge in the design and realization of promising functionalized nanocarbon materials. In the second part, we present scanning probe microscopic characterization of both edge types to experimentally characterize the two electronic states. As predicted, we find the armchair-edged nanographene to have an energetically stable electronic pattern. The zigzag-edged nanographene shows a nonbonding (π radical) pattern, which is the source of the material's electronic and magnetic properties and its chemical activity. Precise control of the edge geometry is a practical requirement to control the electronic structure. We show that we can fabricate the energetically unstable zigzag edges using scanning probe manipulation techniques, and we discuss challenges in using these techniques for that purpose.

Research and discussion on the lock-in threshold variation of ring laser gyro

The lock-in threshold can be obtained by reducing the rotation rate. The lock-in threshold obtained using this method is inconsistent with the angle random walk derived from Allan variance. A dynamic burn-in grating model is proposed. The burn-in effect happens in the mirror not only in the standing wave state, but also in the traveling wave state, and it responds dynamically to beat frequency, which is different from previous research results. It can be derived that the lock-in threshold increases with the decrease of rotation rate for this model, and many puzzling phenomena of the ring laser gyro can be explained.

Impurity-induced quasiparticle interference in the parent compounds of iron-pnictide superconductors

The impurity-induced quasiparticle interference (QPI) in the parent compounds of iron-pnictide superconductors is investigated based on a phenomenological two-orbital four-band model and $T$-matrix method. We find the QPI is sensitive to the value of the magnetic order which may vary from one compound to another. For small values of the magnetic order, the pattern of oscillation in the local density of states (LDOS) induced by the QPI exhibits two-dimensional characteristics, consistent with the standing wave state observed in the 1111 compound. For larger values of the magnetic order, the main feature of the spatial modulation of the LDOS is the existence of one-dimensional stripe structure which is in agreement with the nematic structure in the parent compound of the 122 system. In both cases the system shows ${C}_{2}$ symmetry and only in the larger magnetic order case, there exist in-gap bound states. The corresponding QPI in $q$ space is also presented. The patterns of modulation in the LDOS at nonzero energies are attributed to the interplay between the underlying band structure and Fermi surfaces.

Exact results for the Kuramoto model with a bimodal frequency distribution

We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.

Completeness relation in theR-matrix theory of scattering

We present the biorthogonal complement of the standing wave state and its relation to scattering operators. This establishes a completeness relation in terms of the standing wave states, and therefore completes the $\mathit{R}$-matrix (reactance matrix) theory of scattering. We clarify with an example that the $\mathit{R}$-matrix theory, in its formulation presented here, can claim its due place in addition to the standard $\mathit{T}$-matrix theory.

Manifestation of Work Function Difference in High Order Gundlach Oscillation

Gundlach oscillation (or the standing-wave state) is a general phenomenon manifesting in the tunneling spectrum acquired from a metal surface using scanning tunneling spectroscopy. Previous studies relate the energy shift between peaks of the lowest-order Gundlach oscillation observed on the thin film and the metal substrate to the difference in their work functions. By observing Gundlach oscillations on Ag/Au(111), Ag/Cu(111), and Co/Cu(111) systems, we demonstrate that the work function difference is not the energy shift of the lowest order but the ones of higher order where a constant energy shift exhibits. Higher order Gundlach oscillations can thus be applied to determine the work function of thin metal films precisely.

Instabilities in a One-Dimensional Driven Bistable System under Delayed Feedback Control

We consider the dynamics of a delayed feedback controlled system, which is designed to stabilize an unstable homogeneous symmetric periodic state in a periodically driven bistable system in one dimension. The existence of a finite limit on our ability to measure spatial structure in the feedback system is taken into account by employing Fourier space filtering with a cutoff, which corresponds to the measurement resolution, in the wave number domain. The main purpose of this paper is to clarify the influences of the resolution on the limitation of control and the dynamics in various unstable regimes. First, we give an overview of the dynamics. The characteristic features of the dynamics due to the first and second instabilities, the dynamics in resonant regimes, and the dynamics in a chaotic regime are elucidated. Second, on the basis of a linear stability analysis, we estimate the stability criterion in a conventional control model and an extended control model. Third, we treat the transition from a standing wave state to a vibrating standing wave state in which the mean level of the standing wave undergoes a Hopf bifurcation, and obtain the borders of the transitions for the two controlled models.

Karhunen-Loève analysis of spatiotemporal flame patterns

The ability of Karhunen-Lo\`eve (KL) decomposition to identify, extract, and separate the spatial features that characterize a spatiotemporal system is demonstrated using video images from a combustion experiment and nonstationary states from a phenomenological model. Cellular flames on a circular porous plug burner exhibit a variety of stationary and nonstationary patterns. KL decomposition is used to analyze the spatiotemporal dynamics of four experimental states: one- and two-cell rotating states, two counterrotating rings, a standing-wave state, and two one-cell rotating states from numerical simulations of a phenomenological model designed to study pattern formation in a circular domain. The KL technique optimally captures the dynamics of the states by producing a linear subspace on which the reconstructed dynamics has a minimum truncation error. It identifies the dominant spatial structures whose coupling produces the observed patterns and distinguishes between uniform and nonuniform rotational motion. The implementation of this technique using video images as input is explained and the implications of symmetry in interpreting the KL analysis of the dynamics are described.

Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface Waves.

A two dimensional model is introduced to study pattern formation, secondary instabilities and the transition to spatiotemporal chaos (weak turbulence) in parametric surface waves. The stability of a periodic standing wave state above onset is studied against Eckhaus, zig-zag and transverse amplitude modulations (TAM) as a function of the control parameter $\varepsilon$ and the detuning. A mechanism leading to a finite threshold for the TAM instability is identified. Numerical solutions of the model are in agreement with the stability diagram, and also reveal the existence of a transition to spatiotemporal chaotic states at a finite $\varepsilon$. Power spectra of temporal fluctuations in the chaotic state are broadband, decaying as a power law of the frequency $\omega^{-z}$ with $z \approx 4.0$.

An efficient microwave power source: The free-electron laser afterburner

A kind of microwave power source, called a free-electron laser (FEL) afterburner, that consists of a free-electron laser buncher and a slow-wave output structure sharing a magnetic wiggler field with the buncher is proposed. The buncher and the slow-wave structure can operate in either a traveling-wave state or a standing-wave state. In the buncher, the wiggler field together with the radiation field makes an electron beam bunched, and in the slow-wave structure the wiggler field keeps the beam bunched while the bunched beam interacts strongly with the slow-wave structure and thus produces rf power. The bunching process comes from the free-electron laser mechanism, and the generating process of rf power is in a slow-wave structure. A three-dimensional, time-dependent code is used to simulate a particular standing-wave FEL afterburner and it is shown that rf power of up to 1.4 GW at 17.12 GHz, can be obtained from a 1 kA, 5 MeV electron beam with an energy spread of less than 1% and an emittance of less than 0.5×10−3 π rad m.

Loss Lock-In in the Ring Laser

In the ring laser, frequency synchronization between the oppositely directed traveling waves is usually attributed to mutual coupling by the mechanism of the backscattering of energy from each of the beams into the direction of the other. A new type of coupling is introduced where the standing wave state is shown to have less losses than the traveling wave state for rotations rates less than some critical lock-in threshold. The coupling mechanism is spacially varying losses over the cavity length. The minimum loss condition is also shown to determine the location of the nodes of the standing waves. In certain cases, lock-in thresholds due to nonuniform losses are shown to have the potential of being a few orders-of-magnitude greater than thresholds attributed to backscattering.

Gravitational waves in general relativity II. The reflexion of cylindrical waves

The reflexion of outward travelling cylindrical gravitational waves by a thin cylindrical shell of matter is discussed. The shell is regarded as a surface distribution, and it is shown that if it is perfectly rigid, total reflexion occurs. Quasi-periodic incident waves are found to lead to a completely periodic standing wave state between the reflector and a central wave-source, the field outside the system remaining static. In the treatment of standing waves the components of the stress-energy tensor in the shell are determined and the construction for a model source is given.