Frequency Distribution Of Oscillators Research Articles

Elastic anomalies in glasses: Elastic string theory understanding of the cases of glycerol and silica

We present an implementation of the analytical string theory recently applied to the description of glasses. These are modeled as continuum media with embedded elastic string heterogeneities, randomly located and randomly oriented, which oscillate around a straight equilibrium position with a fundamental frequency depending on their length. The existence of a length distribution reflects then in a distribution of oscillation frequencies which is responsible for the Boson Peak in the glass density of states. Previously, it has been shown that such a description can account for the elastic anomalies reported at frequencies comparable with the Boson Peak. Here we start from the generalized hydrodynamics to determine the dynamic correlation function $S(k,\omega)$ associated with the coherent, dispersive and attenuated, sound waves resulting from a sound-string interference. Once the vibrational density of states has been measured, we can use it for univocally fixing the string length distribution inherent to a given glass. The density-density correlation function obtained using such distribution is strongly constrained, and able to account for the experimental data collected on two prototypical glasses: glycerol and silica. The obtained string length distribution is compatible with the typical size of elastic heterogeneities previously reported for silica and supercooled liquids, and the atomic motion associated to the string dynamics is consistent with the soft modes recently identified in large scale numerical simulations as non-phonon modes responsible for the Boson Peak. The theory is thus in agreement with the most recent advances in the understanding of the glass specific dynamics and offers an appealing simple understanding of the microscopic origin of the latter, while raising new questions on the universality or material-specificity of the string distribution properties.

Analytical statistical study of beam‐pointing error in two‐dimensional coupled oscillator arrays

In this study, the authors investigate the effect of the random free-running frequency distribution of oscillators in two-dimensional (2D) coupled oscillator arrays (COAs) on the phase error of the elements, and consequently on the beam-pointing angle of the array. They derive an analytic formulation for the standard deviation of the beam-pointing error (BPE) in terms of the statistical parameters of the free-running frequencies of the elements and array configuration. The results show that in order to decrease the BPE in 2D COAs, one can use COAs with a larger size (greater number of elements) and decrease the deviation of the free-running frequency of the oscillators.

Community Structure and Multi-Modal Oscillations in Complex Networks

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.

One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes

We study synchronization in a system of phase-only oscillators residing on the sites of a one-dimensional periodic lattice. The oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size-dependent constant. The exponent α of the power law is taken in the range 0≤α<1. The oscillator frequency distribution is symmetric about its mean (taken to be zero) and is nonincreasing on [0,∞). In the continuum limit, the local density of oscillators evolves in time following the continuity equation that expresses the conservation of the number of oscillators of each frequency under the dynamics. This equation admits as a stationary solution the unsynchronized state uniform both in phase and over the space of the lattice. We perform a linear stability analysis of this state to show that when it is unstable, different spatial Fourier modes of fluctuations have different stability thresholds beyond which they grow exponentially in time with rates that depend on the Fourier modes. However, numerical simulations show that at long times all the nonzero Fourier modes decay in time, while only the zero Fourier mode (i.e., the "mean-field" mode) grows in time, thereby dominating the instability process and driving the system to a synchronized state. Our theoretical analysis is supported by extensive numerical simulations.

Low dimensional behavior of large systems of globally coupled oscillators

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

Investigation of the locking bandwidth in linear and circular arrays of mutually coupled oscillators, intended for microwave power combining

An array of N weakly coupled oscillators is investigated in two major configurations. The locking bandwidth for linear and circular arrays is assessed through an array coupling matrix, in a worst-case free-running frequency distribution of oscillators. The superior locking bandwidth for circular arrays is demonstrated.

On the Vlasov Limit for Systems of Nonlinearly Coupled Oscillators without Noise

We discuss the application to systems of coupled nonlinear oscillators of methods originally developed for the Vlasov-Poisson equations of plasma kinetic theory. These methods lead to a simple but rigorous derivation of the infinite-oscillator limit for the well-known Kuramoto model, and also to a compact proof of global existence and uniqueness of solutions to the resulting Kuramoto-Sakaguchi equations. Vlasov-limit techniques can also be adapted to prove a central limit theorem for statistical ensembles of systems of oscillators. However, we emphasize that the noiseless Kuramoto model itself is strictly deterministic, and that the Kuramoto-Sakaguchi equations can be formulated without any reference to a stochastic distribution of oscillator frequencies.

Microscopic Origin of the Optical Kerr Effect Response of CS2−Pentane Binary Mixtures

The femtosecond optically heterodyned detected optical Kerr effect (OHD-OKE) response of carbon disulfide/pentane binary mixtures is reported for various concentrations. As pentane is added, three trends are observed: the instantaneous electronic contribution gets more pronounced, the picosecond diffusive reorientation becomes faster, and the subpicosecond inertia-limited response slows down. The corresponding frequency domain representation of the latter component narrows and shifts toward lower frequencies. These spectral changes occur exclusively when the CS2 concentration is lowered from 100 to 20 vol %; at still lower concentrations no further changes are observed. When the nondiffusive part of the optical response is assigned exclusively to coherent intermolecular vibrations, the spectral changes can be attributed to a reduced inhomogeneous broadening, reflecting a narrowing of the distribution of oscillator frequencies. However, we argue here that because of the observed concentration dependence the validity of this model is questionable. Long-range interaction-induced effects and short-range collision-induced processes are additional scattering mechanisms that may contribute substantially to the observed spectra.

Asymptotic description of transients and synchronized states of globally coupled oscillators

A two-timescale asymptotic method has been introduced to analyze the multimodal mean-field Kuramoto model of oscillator synchronization in the high-frequency limit. The method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution. Each component evolves towards a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them. Synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values. Our results agree very well with direct numerical simulations of the nonlinear Fokker-Planck equation for the probability density. Numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal (symmetric and asymmetric) frequency distribution with or without external field. We also recover in a very easy and intuitive way the only other known analytical results: those corresponding to reflection-symmetric bimodal frequency distributions near bifurcation points.

Loss of frequencies in autocorrelations and a procedure to recover them

We show that the standard methods, like the autocorrelation of a discrete time series, such as a spike train, can often miss prominent oscillation frequencies. In particular, if a phenomenon involves bursting and oscillations at two or more frequencies, the higher frequencies, with a lesser number of spikes per burst, can be often missed by the conventional methods. It is argued that one should separate the intraburst phenomena from the interburst phenomena by identifying bursts as single events. The distribution of oscillation frequencies should then computed by evaluating the autocorrelation of these events. A method is also developed for situations where intraburst and interburst time scales overlap significantly. It is shown that the correct estimate of strengths of various frequencies is found by using different thresholds for burst identification. The method is applied to data from a neuron in the subthalamic nucleus of a Parkinsonian (MPTP-treated) monkey. The conventional analysis shows that the neuron oscillates at only 6 Hz, whereas the new analysis reveals the presence of an additional, predominant, oscillation frequency of 18 Hz.

Optical properties of amorphous compounds As 2S 3 and As 2Se 3 in the infrared region

Optical properties of amorphous As 2S 3 and As 2Se 3 at room temperature in the region between 30 and 8000 cm −1 are described. Absorption peaks at 38 cm −1 for both As 2S 3 and As 2Se 3 are assigned as vibration modes v AS concerning with only arsenic atoms. Kramers-Kronig analysis of reflectivity indicates only one reststrahlen band at 301 cm −1 for As 2S 3 or at 226 cm −1 for As 2Se 3 which is assigned as AsS or AsSe stretching vibration v R. Bands at 698 and 991 cm −1 for As 2S 3 or 487 and 732 cm −1 for As 2Se 3 are interpreted as overtones of v R. The shape of the reststrahlen bands fits to a Gaussian form, and broadening of the band can be attributed to a Gaussian distribution of oscillator frequencies. It is found from the value of the force constants (1.18 and 1.19 md/Å for As 2Se 3 for the stretching vibration and the difference between low and high frequency dielectric constants (1.63 and 1.92 for As 2S 3 and As 2Se 3) that the compounds have a large degree of covalency.