Chain Of Oscillators Research Articles

Higher-order and long-range synchronization effects for classification and computing in oscillator-based spiking neural networks

In the circuit of two thermally coupled VO2 oscillators, we studied a higher order synchronization effect, which can be used in object classification techniques to increase the number of possible synchronous states of the oscillator system. We developed the phase-locking estimation method to determine the values of subharmonic ratio and synchronization effectiveness. In our experiment, the number of possible synchronous states of the oscillator system was twelve, and subharmonic ratio distributions were shaped as Arnold's tongues. In the model, the number of states may reach the maximum value of 150 at certain levels of coupling strength and noise. The long-range synchronization effect in a one-dimensional chain of oscillators occurs even at low values of synchronization effectiveness for intermediate links. We demonstrate a technique for storing and recognizing vector images, which can used for reservoir computing. In addition, we present the implementation of analog operation of multiplication, the synchronization based logic for binary computations, and the possibility to develop the interface between spike neural network and a computer. Based on the universal physical effects, the high order synchronization can be applied to any spiking oscillators with any coupling type, enhancing the practical value of the presented results to expand spike neural network capabilities.

Energy transmission in nonlinear chains of harmonic oscillators with long-range interactions

In this work, the phenomenon of supratransmission in nonlinear sine-Gordon chains with global interactions will be predicted analytically for the first time in the literature. The model considered is an extension of the classical sine-Gordon chain which describes the dynamics of pendula attached through strings. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process for relatively high degrees of interaction. The threshold for the classical model is a particular case of our predictions when the degree of interaction tends to infinity. We confirm numerically the validity of our results using a systematic computational approach. We obtain numerically bifurcation diagrams for the supratransmission threshold and compare them with the theoretical predictions, obtaining a good agreement.

Thermal rectification in one-dimensional lattices with nonlinear system–reservoir coupling

In this work we study the thermal rectification properties of a one-dimensional, mass-graded oscillator lattice with nonlinear system–reservoir couplings. For the case of a harmonic oscillator chain the system presents no rectification at high temperatures and a very weak one for low temperatures as the asymmetric, nonlinear interaction with the thermal reservoirs is increased. The observed rectification rapidly degrades as either the thermal bias or mass asymmetry increase in magnitude. For an anharmonic oscillator Fermi–Pasta–Ulam lattice the rectification increases for the aforementioned conditions, and is maximized when only nonlinear system-reservoir interactions are considered. These latter ones correspond to the case of multiplicative noise affecting the boundaries of the lattice.

Synchronization in linear chains of non-identical vortex-based spin-torque oscillators

Arrays of interacting oscillators having different individual frequencies can exhibit transitions to synchronization depending on array and individual oscillator parameters. In particular, vortex-based spin-torque nanopillar oscillators have gyrotropic frequencies depending inversely on the free layer disk radius. Here the effect of random distribution of radii on synchronization for chains of oscillators is investigated, since in actual systems of spin torque oscillator’s radii will be non-identical owing to fabrication control limitations. The vortex dynamics of linear chains interacting through the dipolar interaction are modeled by the Thiele equations with the free layer radii given by a random distribution about a mean. It is shown that there are two critical current densities for each chain: first, the onset of synchronized gyrotropic motion of the largest radius free layer oscillator, and second, the loss of synchronization as the individual oscillator amplitudes increase. The second critical current density is strongly dependent on the free layer radii of the particular chain. However, the synchronized frequency only weakly depends on the mean radius of each chain, and it is shown to be practically independent of the radii distribution.

Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping

We study the one-dimensional nearest neighbor tight binding model of electrons with independently distributed random hopping and no on-site potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to sub-lattice symmetry, for each realization). For non-singular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\rho(E) \sim 1/|E \ln^3|E||$ and of the localization length $\xi(E) \sim |\ln|E||$, near the band center $E = 0$. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain). Simultaneously, the state at $E = 0$ shows a universal, sub-exponential decay at large distances $\sim \exp [ -\sqrt{r/r_0} ]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\sim 1/ [t \ln^{\lambda+1}(1/t) ]$, characterized by a single, continuously tunable parameter $\lambda > 0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\lambda > 2$, it no longer holds in the interval $0 < \lambda < 2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the non-universal parameter $\lambda$; simultaneously, the localization length shows a less divergent form at low energies, and ceases to diverge below $\lambda = 1$. For $\lambda < 2$, the fall-off of the $E = 0$ state at large distances also deviates from the universal result, and is of the form $\sim \exp [-(r/r_0)^{1/\lambda}]$, which decays faster than an exponential for $\lambda < 1$.

High-heat-flux rectification due to a localized thermal diode.

A theoretical implementation of a localized thermal diode with a rectification factor greater than 10^{6} is demonstrated. In reverse thermal bias, extremely low thermal conductivity is achieved through phononic Rayleigh scattering from a finite-depth defect. In forward bias, the diode oscillator escapes the defect and thermal conductivity becomes up to four orders of magnitude higher. The setup provides a minimal model of a localized thermal diode between two identical oscillator chains and opens up a pathway for thermal diode implementations.

Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N^{-3}. For the purely harmonic chain the order of the rate is in [N^{-3},N^{-1}]. This shows that, in this set up, the spectral gap decays at most polynomially with N.

Global Stabilization of Discrete-Time Linear Systems Subject to Input Saturation and Time Delay

This article studies the problem of global stabilization of discrete-time linear systems subject to input saturation and time delay. The considered time-delay systems are first transformed into delay-free systems based on prediction technique. Then, by utilizing saturation functions technique, the corresponding global stabilizing controllers are proposed for two special discrete-time linear systems-a chain of integrators and oscillators, and explicit conditions guaranteeing stability are also given. Both current and delayed feedback information are utilized in the controller design, and some free parameters are also introduced into these controllers. These advantages can help improve the control performance significantly. Subsequently, a systematic control design procedure for globally stabilizing general discrete-time linear systems subject to multiple inputs and/or multiple inputs delays is proposed. The design procedure is in an explicit and recursive way with explicit conditions guaranteeing stability being given, and thus is easier to use than the existing one. Finally, the effectiveness of the proposed approaches are illustrated by three numerical examples.

Solitary phase waves in a chain of autonomous oscillators.

In the present paper, we study phase waves of self-sustained oscillators with a nearest-neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics, we approximate the lattice with a quasi-continuum (QC). The resulting partial differential model is then further reduced to the Gardner equation, which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations, we determine the shapes of solitary waves, kinks, and the flat-like solitons that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all, we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.

Freak chimera states in a locally coupled Duffing oscillators chain

Arrays of oscillators driven out-of-equilibrium can support the coexistence between coherent and incoherent domains that have become known as chimera states. Recently, we have reported such an intriguing self-organization phenomenon in a chain of locally coupled Duffing oscillators. Based on this prototype model, we reveal a generalization of chimera states corresponding to the coexistence of incoherent domains. These freak states emerge through a bifurcation in which the coherent domain of an existing chimera state experiences an instability giving rise to another incoherent state. Using Lyapunov exponents and Fourier analysis allows us to characterize the dynamical nature of these extended solutions. Taking the Kuramoto order parameter, we were able to compute the bifurcation diagram of freak chimera states.

High Frequency Limit for a Chain of Harmonic Oscillators with a Point Langevin Thermostat

We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin and energy, momentum and volume conserving noise that models the collisions between atoms. The noise is rarefied in the limit, {that corresponds to the hypothesis} that in the macroscopic unit time only a finite number of collisions takes place (Boltzmann-Grad limit). We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function $W(t, y, k)$ that evolves according to a linear kinetic equation, with the interface condition at $y=0$ that corresponds to reflection, transmission and absorption of phonons. The paper extends the results of [3], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.

Stabilization of Statistical Solutions for an Infinite Inhomogeneous Chain of Harmonic Oscillators

Рассматривается бесконечная неоднородная гармоническая цепочка частиц с различными силовыми константами взаимодействия между ними. Изучается поведение при больших временах распределений решений задачи Коши со случайными начальными условиями. Основной результат работы - доказательство сходимости этих распределений к некоторой предельной мере.

Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit

We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling limit. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions. Furthermore, we prove that such linearized evolution holds in some time scales larger than the hyperbolic one.

Fractional diffusion limit for a kinetic equation with an interface

We consider the limit of a linear kinetic equation with reflection-transmission-absorption at an interface and with a degenerate scattering kernel. The equation arises from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected or transmitted upon crossing the interface.

Spatial heterogeneity may form an inverse camel shaped Arnol'd tongue in parametrically forced oscillations.

Frequency locking in forced oscillatory systems typically organizes in "V"-shaped domains in the plane spanned by the forcing frequency and amplitude, the so-called Arnol'd tongues. Here, we show that if the medium is spatially extended and monotonically heterogeneous, e.g., through spatially dependent natural frequency, the resonance tongues can also display "U" and "W" shapes; we refer to the latter as an "inverse camel" shape. We study the generic forced complex Ginzburg-Landau equation for damped oscillations under parametric forcing and, using linear stability analysis and numerical simulations, uncover the mechanisms that lead to these distinct resonance shapes. Additionally, we study the effects of discretization by exploring frequency locking of oscillator chains. Since we study a normal-form equation, the results are model-independent near the onset of oscillations and, therefore, applicable to inherently heterogeneous systems in general, such as the cochlea. The results are also applicable to controlling technological performances in various contexts, such as arrays of mechanical resonators, catalytic surface reactions, and nonlinear optics.

Stabilization of Statistical Solutions for an Infinite Inhomogeneous Chain of Harmonic Oscillators

An infinite inhomogeneous harmonic chain of particles with different force constants of interaction is considered. The large time behavior of distributions of the solutions to the Cauchy problem with random initial data is studied. The main result of the paper establishes the convergence of these distributions to a limiting measure.

Dynamic fracture effects observed in a one-dimensional discrete mechanical system

Dynamic fracture of a one-dimensional chain of identical linear oscillators (masses connected by springs) is considered in the work. The system is supposed to consist of arbitrary but finite number of links and the first mass is supposed to be fixed. Two loading conditions are discussed: free oscillations of an initially statically prestressed chain and loading the chain with a short deformation pulse. Both problems are solved analytically for an arbitrary number of links. The obtained solutions are investigated and a dynamic fracture effect related to an explicitly discrete structure of the system is revealed: a deformation wave travelling through the chain is distorted and some links may be subjected to critical deformation. The obtained solutions for the chain are compared to the solutions of analogous problems stated for an elastic rod – a continuum counterpart of the considered discrete system. It is shown that the discussed fracture effect cannot be found in a continuous system.

Мобильные диссипативные бризеры в цепочке нелинейных осцилляторов

In a numerical experiment, it was found that in a chain of Raleigh oscillators with a nonlinear connection between them, at least three localized stationary breathers modes may exist: non-mobile, "slow" and "fast" dissipative breathers. The dynamics of the chain elements depends on the ratio of characteristic time scales of the system which determined by the frequency of oscillators and the rigidity of the connection between them. Considered system is multistable.

Frequency Multiplier and Mixer MMICs Based on a Metamorphic HEMT Technology Including Schottky Diodes

This paper reports on the monolithic integration of layout-optimized Schottky diodes realized in an established 50-nm gate-length metamorphic high-electron-mobility transistor technology for use in multifunctional nonlinear circuits. The suitability of the realized Schottky diodes is demonstrated by a broadband millimeter-wave I/Q-mixer (In-phase/Quadrature) and local oscillator (LO) chain comprising two power amplifiers and a frequency tripler, fabricated on monolithic microwave integrated circuits (MMICs). Both circuits are based on an anti-parallel Schottky diode topology. The subharmonically-pumped I/Q-mixer covers an RF (radio frequency) and IF (intermediate frequency) range of at least 75 GHz to 110 GHz and 0.5 GHz to 15 GHz, respectively. The single-sideband conversion loss is between 14 dB and 16 dB across most of the entire RF and IF bands. The core of the LO chain consists of a frequency tripler (multiplier by three) and features a bias-adjustable output power with almost constant conversion efficiency and a control range of more than 8 dB. The fully-integrated LO chain MMIC matches the needs of the presented I/Q-mixer and exhibits an average output power of 16.3 dBm with a covered frequency range of 38 GHz to 60 GHz. The unwanted harmonics are suppressed by at least -25.9 dBc below the third harmonic for the entire frequency range and better than -32.1 dBc for most part of the band. Thus, the mixer and tripler MMICs demonstrate state-of-the-art performance with regards to, e.g., covered bandwidth, output power, harmonic suppression, or 1 dB compression point.

Nonlinear strain waves in a metamaterial defined a mass-to-mass chain

A mathematical model is proposed that is a chain of oscillators consisting of nonlinear elastic elements and masses, each of which contains an internal nonlinear oscillator. Such a model describes a mass-to-mass acoustic metamaterial class. It is shown that the obtained system of equations can be reduced to the nonlinear evolutionary equation of Benjamin-Bona-Mahoni, showing that spatially localized nonlinear deformation waves (solitons) can form in the metamaterial under dynamic action on it.