Classical Oscillator Research Articles

Predicting phase and sensing phase coherence in chaotic systems with machine learning.

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

Equilibrium stationary coherence in the multilevel spin-boson model

Interaction between a quantum system and its environment can induce stationary coherences -- off-diagonal elements in the reduced system density matrix -- even at equilibrium. This work investigates the ``quantumness'' of such phenomena by examining the ability of classical and semiclassical models to describe equilibrium stationary coherence in the multi-level spin boson (MLSB) model, a common model for light-harvesting systems. A well justified classical harmonic oscillator model is found to fail to capture equilibrium coherence. This failure is attributed to the effective weakness of classical system-bath interactions due to the absence of a discrete system energy spectrum and, consequently, of quantized shifts in oscillator coordinates. Semiclassical coherences also vanish for a dimeric model with parameters typical of biological light-harvesting, i.e., where both system sites couple to the bath with the same reorganization energy. In contrast, equilibrium coherence persists in a fully quantum description of the same system, suggesting a uniquely quantum-mechanical origin for equilibrium stationary coherence in, e.g., photosynthetic systems. Finally, as a computational tool, a perturbative expansion is introduced that, at third order in $\hbar$, gives qualitatively correct behavior at ambient temperatures for all configurations examined.

Sachdev-Ye-Kitaev superconductivity: Quantum Kuramoto and generalized Richardson models

Sachdev-Ye-Kitaev (SYK) model has emerged as a new paradigm of the non-Fermi-liquid behavior. Here we investigate a possibility of having a superconducting off-diagonal long-range order (ODLRO) and a pseudogap phase within the SYK framework. We found that ODLRO may be established in spin-1/2 version of the model with the time-reversal invariance and an extra attractive interaction. If the latter is taken as the on-site negative-$U$ Hubbard term, it leads to the pseudogap phase at $U<U_c$ dominated by quantum fluctuations of local phases. These fluctuations are described by a quantum version of the Kuramoto model, traditionally employed to illustrate synchronization of classical non-linear oscillators. In the opposite limit of large $U$, the SYK+Hubbard model is approaching a certain generalization of the integrable Richardson model. We present exact diagonalization studies, along with analytic solutions of the aforementioned limiting cases. We also discuss possible holographic interpretations of the model, ODLRO and the pseudogap.

Quantum superexponential oscillator

We investigate the spectral and eigenstate properties of the quantum superexponential oscillator. Our focus is on the quantum signatures of the recently observed transition of the energy dependent period of the corresponding classical superexponential oscillator. We show that the ground state exhibits a remarkable metamorphosis of decentering, asymmetrical squeezing and the development of a tail. Analyzing the central moments up to high order a characteristic transition from exponentially decaying moments to increasing moments is unravelled. A corresponding spectral analysis shows that, surprisingly, to a good approximation the spectrum is equidistant. A closer look, however, reveals a spectral scaling behaviour below the transition point which is replaced by irregular oscillations above the transition energy. Excited bound states are analyzed up to the continuum threshold. We discuss future perspectives and possible experimental realizations of the superexponential oscillator.

Optical response of atom chains beyond the limit of low light intensity: The validity of the linear classical oscillator model

Atoms subject to weak coherent incident light can be treated as coupled classical linear oscillators, supporting subradiant and superradiant collective excitation eigenmodes. We identify the limits of validity of this \emph{linear classical oscillator model} at increasing intensities of the drive by solving the quantum many-body master equation for coherent and incoherent scattering from a chain of trapped atoms. We show that deviations from the linear classical oscillator model depend sensitively on the resonance linewidths $\upsilon_\alpha$ of the collective eigenmodes excited by light, with the intensity at which substantial deviation occurs scaling as a powerlaw of $\upsilon_\alpha$. The linear classical oscillator model then becomes inaccurate at much lower intensities for subradiant collective excitations than superradiant ones, with an example system of seven atoms resulting in critical incident light intensities differing by a factor of 30 between the two cases. By individually exciting eigenmodes we find that this critical intensity has a $\upsilon_\alpha^{2.5}$ scaling for narrower resonances and more strongly interacting systems, while it approaches a $\upsilon_\alpha^3$ scaling for broader resonances and when the dipole-dipole interactions are reduced. The $\upsilon_\alpha^3$ scaling also corresponds to the semiclassical result whereby quantum fluctuations between the atoms have been neglected. We study both the case of perfectly mode-matched drives and the case of standing wave drives, with significant differences between the two cases appearing only at very subradiant modes and positions of Fano resonances.

Vortex-Induced Vibrations of A Free-Spanning Pipe Based on A Nonlinear Hysteretic Soil Model at the Shoulders

The pipe-soil interactions at shoulders can significantly affect the vortex-induced vibrations (VIV) of free-spanning pipes in the subsea. In this paper, the seabed soil reacting force on the pipe is directly calculated with a nonlinear hysteretic soil model. For the VIV in the middle span, a classic van der Pol wake oscillator is adopted. Based on the Euler-Bernoulli beam theory, the vibration equations of the pipe are obtained which are different in the middle span and at the two end shoulders. The static configuration of the pipe is firstly calculated and then the VIV is simulated. The present model is validated with the comparisons of VIV experiment, pipe-soil interaction experiment and the simulation results of VIV of free-spanning pipes in which the seabed soil is modelled with spring-dashpots. With the present model, the influence of seabed soil on the VIV of a free-spanning pipe is analyzed. The parametric studies show that when the seabed soil has a larger suction area, the pipe vibrates with smaller bending stresses and is safer. While with the increase of the shear strength of the seabed soil, the bending stresses increase and the pipe faces more danger.

Prediction of Vortex-Induced Vibration Response of Deep Sea Top-Tensioned Riser in Sheared Flow Considering Parametric Excitations

Abstract It is widely accepted that vortex-induced vibration (VIV) is a major concern in the design of deep sea top-tensioned risers, especially when the riser is subjected to axial parametric excitations. An improved time domain prediction model was proposed in this paper. The prediction model was based on classical van der Pol wake oscillator models, and the impacts of the riser in-line vibration and vessel heave motion were considered. The finite element, Newmark-β and Newton‒Raphson methods were adopted to solve the coupled nonlinear partial differential equations. The entire numerical solution process was realised by a self-developed program based on MATLAB. Comparisons between the numerical calculation and the published experimental test were conducted in this paper. The in-line and cross-flow VIV responses of a real size top-tensioned riser in linear sheared flow were analysed. The effects of the vessel heave amplitude and frequency on the riser VIV were also studied. The results show that the vibration displacements of the riser are larger than the case without vessel heave motion. The vibration modes and frequencies of the riser are also changed due to the vessel heave motion

Temperature Evolution of the Dielectric Response of α-Lactose Monohydrate in the THz Frequency Range

Transmission spectra of pressed microcrystalline α-lactose monohydrate samples have been measured by THz time-domain spectroscopy in the temperature range of existence of the solid phase (10–475 K). An analysis of the observed absorption lines using the model of classical oscillator have revealed complex temperature evolution of resonance eigenfrequencies and determined the range of existence of a phase of α-lactose monohydrate response. The obtained data can be used in practice in various fields of THz optics, including pharmacology, food industry, analytical chemistry, and biophotonics.

Modelling equilibration of local many-body quantum systems by random graph ensembles

We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an excellent proxy for equilibration times. Since maximum flow values are less expensive to compute, they give access to approximate equilibration times for system sizes beyond those accessible by exact diagonalisation.

Scalars Gliding through an Expanding Universe.

In this Letter, we investigate the effects of single derivative mixing in massive bosonic fields. In the regime of large mixing, we show that this leads to striking changes of the field dynamics, delaying the onset of classical oscillations and decreasing, or even eliminating, the friction due to Hubble expansion. We highlight this phenomenon with a few examples. In the first example, we show how an axionlike particle can have its number abundance parametrically enhanced. In the second example, we demonstrate that the QCD axion can have its number abundance enhanced allowing for misalignment driven axion dark matter all the way down to f_{a} of order astrophysical bounds. In the third example, we show that the delayed oscillation of the scalar field can also sustain a period of inflation. In the last example, we present a situation where an oscillating scalar field is completely frictionless and does not dilute away in time.

Birefringent and Complex Dielectric Functions of Monolayer WSe2 Derived by Spectroscopic Ellipsometer

Because of its highly anisotropic confinement of its crystal structure, monolayer WSe2 should exhibit a birefringent optical property, i.e., the light of in-plane and vertical polarization will experience different velocity and loss when propagating in the vicinity of such a material. However, most research so far treats monolayer WSe2 as an isotropic material and ignores the important difference upon light–matter interaction at different polarizations. In this work, monolayer WSe2 is studied as a uniaxial crystal with different complex dielectric functions with respect to the in-plane and vertical polarization. The sample is measured by a spectroscopic ellipsometer, followed by a strict modeling process with a series of classic oscillators. The results show that the complex dielectric functions of monolayer WSe2 are clearly polarization-dependent. The obtained data are essential for the design of polarization-dependent integrated optoelectronic devices utilizing monolayer WSe2 as a key functional material.

Floquet engineering in topological crystalline insulator

There is a peculiar type of insulator, which is protected by the crystal symmetry known as topological crystalline insulator (TCI). In off-resonant cases, Floquet theory is an another way to study conventional Rabi oscillations. By using Floquet theory, the various type of Dirac fermionic systems and phases can be distinguished. In this article, it is shown that Floquet frequency can be used as a tool to distinguish different phases of TCI. The study of Bloch-Siegert shift has been performed and shown its variation in different phases of TCI. The collapse-revival spectra have also been studied in the perspective of Floquet theory and shown how quantum and classical Floquet oscillations are related to each other. The verification of the Floquet theory is justified by using numerical simulation.

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.

Emergent Spaces for Coupled Oscillators.

Systems of coupled dynamical units (e.g., oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering “bespoke” coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of “effective” parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible extensions of this approach, including the possibility of obtaining data-driven effective partial differential equations for coarse-grained neuronal network behavior, as illustrated by the synchronization dynamics of Hodgkin–Huxley type neurons with a Chung-Lu network. Thus, we build an integrated suite of tools for obtaining data-driven coarse variables, data-driven effective parameters, and data-driven coarse-grained equations from detailed observations of networks of oscillators.

Squeezing of coherent light coupled to a periodically driven two-photon anharmonic oscillator

The electromagnetic field coupled to a $$(2l-1)$$ order of nonlinear medium leads to the model of a 2lth anharmonic oscillator, $$l\ge 2$$ being an integer. The oscillator with $$l=2$$ corresponds to the model of a quartic anharmonic oscillator. In spite of the huge advancement of mathematical physics, we are still in search of exact analytical solution to the lowest order (i.e., $$l=2)$$ of anharmonic oscillator. There are few interesting reports where the approximate solutions to the classical and quantum quartic oscillator problems are explored. In the present investigation, the exact analytical solutions of the quartic anharmonic oscillator under rotating wave approximation and hence the two-photon anharmonic oscillator with periodic forcing are exhibited. The solutions corresponding to the quantized two-photon anharmonic oscillator with periodic forcing is used to calculate the second-order variances of the canonically conjugate quadrature in terms of the initial coherent state. The effects of the forcing parameters on the second-order variances and hence on the squeezing effects are examined for various photon numbers and the dimensionless interaction constants. In these studies, we assume that the oscillator is in on-resonance with those of the external forcing term. The off-resonant contribution is found to be small and is neglected.

Strong coupling between excitons and guided modes in WS2-based nanostructures

Transition-metal dichalcogenides (TMDCs) exhibit great potential in light–matter interaction due to their semiconducting nature and remarkable excitonic properties. Here a simple guided resonance structure is proposed to numerically investigate and realize dual-band light absorption exploiting the guided mode resonance (GMR) in the system and the robust exciton in the W S 2 monolayer. The dispersive GMR can strongly interact with the nondispersive exciton state by tuning the incident angle, resulting in remarkable Rabi splitting and the emergence of two hybrid polariton bands. The strong GMR–exciton coupling behavior can be well described by the classic oscillator model. The influence of the geometrical parameters on the spectral response is investigated. In addition, we also find that the Rabi splitting is strongly affected by the position of the W S 2 monolayer in the dielectric slab. Our findings provide a simple route for realization of hybrid light–TMDCs interactions, which may inspire related studies on compact, scalable, and easy-to-fabricate TMDC-based devices.

Energy partition for anharmonic, undamped, classical oscillators

Using stochastic methods, general formulas for average kinetic and potential energies for anharmonic, undamped (frictionless), classical oscillators are derived. It is demonstrated that for potentials of |x|ν, (ν > 0) type energies are equipartitioned for the harmonic potential only. For potential wells weaker than parabolic potential energy dominates, while for potentials stronger than parabolic kinetic energy prevails. Due to energy conservation, the variances of kinetic and potential energies are equal. In the limiting case of the infinite rectangular potential well (ν → ∞) the whole energy is stored in the form of the kinetic energy and variances of energy distributions vanish.

Nested mixed-mode oscillations, part II: Experimental and numerical study of a classical Bonhoeffer–van der Pol oscillator

The dynamics of Bonhoeffer–van der Pol (BVP) oscillators are known to be equivalent to those of the FitzHugh–Nagumo model and have been extensively studied for many years. In a previous work (Inaba and Kousaka, 2020), we discovered nested mixed-mode oscillations (MMOs) generated by a piecewise-smooth driven Bonhoeffer–van der Pol oscillator. In this study, we focus on the MMOs that occur between the 1s- and 1s+1-generating regions for s=2 and 3 in a classical BVP oscillator where the nonlinear conductor is expressed as a third-order polynomial function, and we confirm the occurrence of nested mixed-mode oscillation-incrementing bifurcations (MMOIBs) that are at least doubly nested. Simple (un-nested), singly nested, and doubly nested MMOIBs generate [1s,1s+1×n]n+1, [A1,B1×n], and [A2,B2×n] MMO sequences, respectively, for successive n. A1=[1s,1s+1×m]m+1 and B1=[1s,1s+1×(m+1)]m+2 in the singly nested case for integers m and A2=[[1s,1s+1×l]l+1,[1s,1s+1×(l+1)]l+2×m](l+2)m+(l+1) and B2=[[1s,1s+1×l]l+1,[1s,1s+1×(l+1)]l+2×(m+1)](l+2)(m+1)+(l+1) in the doubly nested case for integers l and m. In particular, we numerically confirm that m=1 with s=2 and 3 cases for the singly nested MMOIBs and that l=1,m=1 with s=2 and 3 cases for the doubly nested MMOIBs. We also show that both the simple (un-nested) and nested MMOIB-generated MMOs have asymmetric Farey characteristics. In addition, we find that these numerical results are well-explained by effectively one-dimensional (1D) Poincaré return maps derived numerically from the dynamics of a constrained driven BVP oscillator that includes a diode with grazing–sliding characteristics. Finally, we verify these numerical results for the classical and constrained BVP oscillators in circuit experiments and derive the first return plots and 1D Poincaré return maps based on laboratory measurements.

Oscillation criteria of Kneser-type for second-order half-linear advanced differential equations

In the paper, we provide an extension of the classical Kneser oscillation theorem to a class of second-order half-linear advanced differential equations. Our results are unimprovable in the sense that the involved oscillation constant is best-possible. As far as we know, no similar result is known even in the linear case. In contrast with existing related results, we relax the common assumption that the deviating function is nondecreasing.

Ultra-Low Voltage Push-Pull Converter for Micro Energy Harvesting

Ultra-low power electronic devices, as by the way of example sensor nodes, can be powered by environmental energy harvesting. Thermo generators, solar cells, inductive generators, or radio frequency antennas provide an extremely low output voltage that must be amplified to generate a suitable power supply. This brief presents an autonomous self-starting ultra-low voltage DC-DC converter that starts operations at 9 mV. The circuit is inspired by the classical zero voltage switching push-pull Royer oscillator, suitably modified by using JFETs to operate at low voltages. Thanks to the soft switching of the transistors and to the symmetrical use of the transformer, it can reach high voltage gain, high frequency, and high efficiency. Theoretical and simulation-based optimizations were used to design the converter; the results were experimentally verified with several prototypes.