Chain Of Oscillators Research Articles

Two mechanisms of remote synchronization in a chain of Stuart-Landau oscillators.

Remote synchronization implies that oscillators interacting not directly but via an additional unit (hub) adjust their frequencies and exhibit frequency locking while the hub remains asynchronous. In this paper, we analyze the mechanisms of remote synchrony in a small network of three coupled Stuart-Landau oscillators using recent results on higher-order phase reduction. We analytically demonstrate the role of two factors promoting remote synchrony. These factors are the nonisochronicity of oscillators and the coupling terms appearing in the second-order phase approximation. We show a good correspondence between our theory and numerical results for small and moderate coupling strengths.

A quasi-periodic route to chaos in a parametrically driven nonlinear medium

Small-sized systems exhibit a finite number of routes to chaos. However, in extended systems, not all routes to complex spatiotemporal behavior have been fully explored. Starting from the sine-Gordon model of parametrically driven chain of damped nonlinear oscillators, we investigate a route to spatiotemporal chaos emerging from standing waves. The route from the stationary to the chaotic state proceeds through quasiperiodic dynamics. The standing wave undergoes the onset of oscillatory instability, which subsequently exhibits a different critical frequency, from which the complexity originates. A suitable amplitude equation, valid close to the parametric resonance, makes it possible to produce universe results. The respective phase-space structure and bifurcation diagrams are produced in a numerical form. We characterize the relevant dynamical regimes by means of the largest Lyapunov exponent, the power spectrum, and the evolution of the total intensity of the wave field.

Two decades of INO research on laser arrays

We summarize in this work the salient points of the studies carried out at the National Institute of Optics (INO) in Florence in the vast field of coherent laser arrays. The research began in the early 90s under the push of Tito Arecchi with the study of coupled laser oscillators. These systems stimulated Tito's curiosity by representing an accessible sample of nonlinear oscillator chains. In the following years at INO we continued our research in this field, obtaining numerical and experimental results both with the realization of laser arrays and in the field of their possible application. Of particular importance for obtaining a fruitful application was the development of optical phase filtering techniques aimed at increasing the optical quality of the beam emitted from these sources, which is actually far from having a traditional geometry.

A plentiful supply of soliton solutions for DNA Peyrard–Bishop equation by means of a new auxiliary equation strategy

In this paper, we present a work on dynamic equation of Deoxyribonucleic acid (DNA) derived from the Peyrard–Bishop (PB) model oscillator chain for various dynamical solitary wave solutions. In order to construct novel soliton solutions in the DNA dynamic PB model with beta-derivative, the efficiency of the newly developed algorithms is being investigated, which could include a new auxiliary equation strategy (NAES). Some precise soliton solutions comprising dark, light and other forms of multi-wave soliton solutions are achieved via the proposed methods. Furthermore, mathematical models demonstrate the singularity of our work in comparison to current literary materials and even describe some results using the classic Peyrard–Bishop model. All the established results contribute to the possibility of extending the approach to solve other nonlinear equations of fractional space–time derivatives in nonlinear sciences. The strategy that has been proposed recently is specific and is being employed to produce novel closed-form solutions for all many other FNLEEs.

On the nonlinear wave transmission in a nonlinear continuous hyperbolic regime with Caputo-type temporal fractional derivative

This present manuscript studies a nonlinear hyperbolic model in fractional form which generalizes the nonlinear Klein–Gordon system. The equation under investigation includes the presence of a time-fractional operator of the Caputo type. A space-fractional form of that equation with integer-order temporal derivative has been previously investigated to elucidate the existence of localized wave transmission in relativistic wave equations. Here, we employ numerical techniques to estimate the solution of the fractional equation. The method has consistency of fourth order in space. Meanwhile, the temporal order of consistency is equal to 3−α. We considered herein a sinusoidal perturbation of the medium. The simulations show the existence of the transmission of localized nonlinear modes in some complex fractional media governed by hyperbolic models. Physically, the present work investigates the phenomenon of nonlinear supratransmission in a continuous generalization of linear chains of harmonic oscillators with memory effects. In particular, the present work corroborates the presence of this nonlinear phenomenon in chains of pendula with memory and arrays of Josephson junctions attached through superconducting wires and memory effects.

Controlled Motion of a Linear Chain of Oscillators

Controlled Motion of a Linear Chain of Oscillators

40 GHz VCO and Frequency Divider in 28 nm FD-SOI CMOS Technology for Automotive Radar Sensors

This paper presents a 40 GHz voltage-controlled oscillator (VCO) and frequency divider chain fabricated in STMicroelectronics 28 nm ultrathin body and box (UTBB) fully depleted silicon-on-insulator (FD-SOI) complementary metal-oxide–semiconductor (CMOS) process with eight metal layers back-end-of-line (BEOL) option. VCOs architecture is based on an LC-tank with p-type metal-oxide–semiconductor (PMOS) cross-coupled transistors. VCOs exhibit a tuning range (TR) of 3.5 GHz by exploiting two continuous frequency tuning bands selectable via a single control bit. The measured phase noise (PN) at 38 GHz carrier frequency is −94.3 and −118 dBc/Hz at 1 and 10 MHz frequency offset, respectively. The high-frequency dividers, from 40 to 5 GHz, are made using three static CMOS current-mode logic (CML) Master-Slave D-type Flip-Flop stages. The whole divider factor is 2048. A CMOS toggle flip-flop architecture working at 5 GHz was adopted for low frequency dividers. The power dissipation of the VCO core and frequency divider chain are 18 and 27.8 mW from 1.8 and 1 V supply voltages, respectively. Circuit functionality and performance were proved at three junction temperatures (i.e., −40, 25, and 125 °C) using a thermal chamber.

Cutoff Thermalization for Ornstein\u2013Uhlenbeck Systems with Small L\xe9vy Noise in the Wasserstein Distance

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems (X^varepsilon _t(x))_{tgeqslant 0} with varepsilon -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, alpha -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp infty /0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure mu ^varepsilon along a time window centered on a precise varepsilon -dependent time scale mathfrak {t}_varepsilon . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result mathcal {W}_p(X^varepsilon _{t_varepsilon + r}(x), mu ^varepsilon ) cdot varepsilon ^{-1} rightarrow Kcdot e^{-q r} for any rin mathbb {R} as varepsilon rightarrow 0 for some spectral constants K, q>0 and any pgeqslant 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of mathcal {Q}. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to varepsilon -small Brownian motion or alpha -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Discrete breathers and discrete oscillating kink solution in the mass-in-mass chain in the state of acoustic vacuum

The present study is concerned with the dynamics of special localized solutions emerging in the mass-in-mass anharmonic oscillatory chain in the state of acoustic vacuum. Each outer element of the chain incorporates an additional, purely nonlinear mass attachment. Using the homogeneity of the system under consideration, we use the separation of time and space and reduce the analysis of standing wave solutions supported by the chain to the analysis of an algebraic system. An analytical study of the latter revealed the distinct types of static discrete breather solutions as well as the discrete oscillating kinks. Along with the analytical description of their spatial wave profiles, we also establish their zones of existence in the space of system parameters. The stability properties of these solutions are assessed through the linear stability analysis (Floquet). All analytical models are supported by the numerical simulations of the full model.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Metachronal waves in concentrations of swimming Turbatrix aceti nematodes and an oscillator chain model for their coordinated motions.

At high concentration, free swimming nematodes known as vinegar eels (Turbatrix aceti), collectively exhibit metachronal waves near a boundary. We find that the frequency of the collective traveling wave is lower than that of the freely swimming organisms. We explore models based on a chain of oscillators with nearest-neighbor interactions that inhibit oscillator phase velocity. The phase of each oscillator represents the phase of the motion of the eel's head back and forth about its mean position. A strongly interacting directed chain model mimicking steric repulsion between organisms robustly gives traveling wave states and can approximately match the observed wavelength and oscillation frequency of the observed traveling wave. We predict body shapes assuming that waves propagate down the eel body at a constant speed. The phase oscillator model that impedes eel head overlaps also reduces close interactions throughout the eel bodies.

Soliton solutions for fractional DNA Peyrard-Bishop equation via the extended -expansion method

In this article, the dynamical DNA equation comes into existence in the oscillatory chain, categorized as the Peyrard-Bishop model. The fractional Peyrard-Bishop model has been studied via β-derivative and M-truncated derivative, for obtaining abundant solitary wave solutions. The proposed model is examined via M-truncated derivative for the first time in this article. The traveling wave transformation method is used to convert the proposed model into an ordinary differential equation. The extended -expansion method is applied to extract dark solitons, periodic solutions, singular solitons, combo solitons and rational solutions. On comparing our extracted solutions with the existing solutions in the literature, it is found that our results are new and not found in the literature. The obtained results are also explained graphically by plotting 3D plots to understand the phenomena of the proposed model.

Breather arrest in a chain of damped oscillators with Hertzian contact

Breather propagation in a damped oscillatory chain with Hertzian nearest-neighbour coupling is investigated. The breather propagation exhibits an unusual two-stage pattern. The first stage is characterized by power-law decay of the breather amplitude. This stage extends over finite number of the chain sites. Drastic drop of the breather amplitude towards the end of this finite fragment is referred to as breather arrest. At the second stage, the breather exhibits very small amplitudes with hyper-exponential decay. Numeric results are rationalized by considering a simplified model of two damped linear oscillators coupled by Hertzian contact forces. Initial excitation of one of these oscillators results in a finite number of beating cycles in the system. This simplified model reliably predicts main features of the breather arrest. More general coupling potentials and effect of pre-compression on the breather propagation are also discussed.

Beyond the universal Dyson singularity for 1-D chains with hopping disorder

We study a simple non-interacting nearest neighbor tight-binding model in one dimension with disorder, where the hopping terms are chosen randomly. This model exhibits a well-known singularity at the band center both in the density of states and localization length. If the probability distribution of the hopping terms is well-behaved, then the singularities exhibit universal behavior, the functional form of which was first discovered by Freeman Dyson in the context of a chain of classical harmonic oscillators. We show here that this universal form can be violated in a tunable manner if the hopping elements are chosen from a divergent probability distribution. We also demonstrate a connection between a breakdown of universality in this quantum problem and an analogous scenario in the classical domain — that of random walks and diffusion with anomalous exponents.

Space–Time Statistical Solutions for a Two-Component Chain of Harmonic Oscillators

Space–Time Statistical Solutions for a Two-Component Chain of Harmonic Oscillators

Harmonic chains and the thermal diode effect.

Harmonic oscillator chains connecting two harmonic reservoirs at different constant temperatures cannot act as thermal diodes, irrespective of structural asymmetry. However, here we prove that perfectly harmonic junctions can rectify heat once the reservoirs (described by white Langevin noise) are placed under temperature gradients, which are asymmetric at the two sides, an effect that we term "temperature-gradient harmonic oscillator diodes." This nonlinear diode effect results from the additional constraint-the imposed thermal gradient at the boundaries. We demonstrate the rectification behavior based on the exact analytical formulation of steady-state heat transport in harmonic systems coupled to Langevin baths, which can describe quantum and classical transport, both regimes realizing the diode effect under the involved boundary conditions. Our study shows that asymmetric harmonic systems, such as room-temperature hydrocarbon molecules with varying side groups and end groups, or a linear lattice of trapped ions may rectify heat by going beyond simple boundary conditions.

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices.

The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi etal. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955 (unpublished)]. We introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi etal. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.

Thermal conduction in a harmonic chain coupled to two cavity-optomechanical systems

We propose a model including a one-dimensional harmonic chain of oscillators whose two ends coupled two cavity-optomechanical systems for studying one-dimensional thermal conductivity in statistical physics. In this model, the cavity-optomechanical systems function as two laser-engineerable thermal reservoirs. When the effective temperatures of the two reservoirs are not equal, a heat flux through the chain can be generated, and moreover the classical and quantum features of the heat flux have been studied. We further show that the heat flux does not obey the Fourier's law in this model. The thermal switching phenomenon could be induced by controlling the on-site vibrational frequency. Finally, we propose to apply the correlation between two mechanical oscillators to measure the heat flux through the chain.

Changes in geometrical aspects of a simple model of cilia synchronization control the dynamical state, a possible mechanism for switching of swimming gaits in microswimmers.

Active oscillators, with purely hydrodynamic coupling, are useful simple models to understand various aspects of motile cilia synchronization. Motile cilia are used by microorganisms to swim and to control the flow fields in their surroundings; the patterns observed in cilia carpets can be remarkably complex, and can be changed over time by the organism. It is often not known to what extent the coupling between cilia is due to just hydrodynamic forces, and neither is it known if it is biological or physical triggers that can change the dynamical collective state. Here we treat this question from a very simplified point of view. We describe three possible mechanisms that enable a switch in the dynamical state, in a simple scenario of a chain of oscillators. We find that shape-change provides the most consistent strategy to control collective dynamics, but also imposing small changes in frequency produces some unique stable states. Demonstrating these effects in the abstract minimal model proves that these could be possible explanations for gait switching seen in ciliated micro organisms like Paramecium and others. Microorganisms with many cilia could in principle be taking advantage of hydrodynamic coupling, to switch their swimming gait through either a shape change that manifests in decreased coupling between groups of cilia, or alterations to the beat style of a small subset of the cilia.

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t^{-frac{1}{3}} for position and momentum correlations and as t^{-frac{2}{3}} for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t^{-frac{1}{4}} for position and momentum correlators and with rate t^{-frac{1}{2}} for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.