System Of Coupled Oscillators Research Articles

Analytical Solution to the Generalized Complex Duffing Equation.

Future scientific and technological evolution in many areas of applied mathematics and modern physics will necessarily depend on dealing with complex systems. Such systems are complex in both their composition and behavior, namely, dealing with complex dynamical systems using different types of Duffing equations, such as real Duffing equations and complex Duffing equations. In this paper, we derive an analytical solution to a complex Duffing equation. We extend the Krýlov-Bogoliúbov-Mitropólsky method for solving a coupled system of nonlinear oscillators and apply it to solve a generalized form of a complex Duffing equation.

Synchronization dynamics of phase oscillator populations with generalized heterogeneous coupling

The Kuramoto model, serving as a paradigmatic tool, has been used to shed light on the collective behaviors in large ensembles of coupled dynamic agents. It is well known that the model displays a second-order(continuous) phase transition towards synchrony by increasing the homogeneous(uniform) global coupling strength. Recently, there is a great interest in investigating the effects of the heterogeneous patterns on the collective dynamics of coupled oscillator systems. Here, we consider a generalized Kuramoto model of globally coupled phase oscillators with quenched disorder in their natural frequencies and coupling strength. By correlating these two types of inhomogeneity, we systematically explore the impacts of heterogeneous structure, the correlation exponent, and the intrinsic frequency distribution on the synchronized dynamics. We develop an analytic framework for capturing the essential dynamic properties involved in synchronization transition. In particular, we demonstrate that the forward critical threshold describing the onset of synchronization is unaffected by the location of heterogeneity, which, however, does depend crucially on the correlation exponent and the form of frequency distribution. Furthermore, we reveal that the backward critical points featuring the desynchronization transition are significantly shaped by all the considered effects, thereby enhancing or weakening the ability of synchronization transition (bifurcation). Our investigation is a step forward in highlighting the importance of heterogeneous pattern presented in the complex systems, and could, thus, provide significant insights for designing the strategy of inducing and controlling synchronization.

KAM Tori for the System of Coupled Quantum Harmonic Oscillators with Reversible Perturbations

KAM Tori for the System of Coupled Quantum Harmonic Oscillators with Reversible Perturbations

Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

AbstractWe deal with the following system of coupled asymmetric oscillators $$\begin{aligned} \left\{ \begin{array}{l} \ddot{x}_1+a_1x_1^+-b_1x^-_1+\phi _1(x_2)=p_1(t) \\ \ddot{x}_2+a_2\,x_2^+-b_2\,x^-_2+\phi _2(x_1)=p_2(t), \end{array} \right. \end{aligned}$$ x ¨ 1 + a 1 x 1 + - b 1 x 1 - + ϕ 1 ( x 2 ) = p 1 ( t ) x ¨ 2 + a 2 x 2 + - b 2 x 2 - + ϕ 2 ( x 1 ) = p 2 ( t ) , where $$\phi _i: \mathbb {R} \rightarrow \mathbb {R}$$ ϕ i : R → R is locally Lipschitz continuous and bounded, $$p_i: \mathbb {R} \rightarrow \mathbb {R}$$ p i : R → R is continuous and $$2\pi $$ 2 π -periodic and the positive real numbers $$a_i, b_i$$ a i , b i satisfy $$\begin{aligned} \dfrac{1}{\sqrt{a_i}}+\dfrac{1}{\sqrt{b_i}}=\dfrac{2}{n}, \quad \text{ for } \text{ some } n \in \mathbb {N}. \end{aligned}$$ 1 a i + 1 b i = 2 n , for some n ∈ N . We define a suitable function $$L: \mathbb {T}^2 \rightarrow \mathbb {R}^2$$ L : T 2 → R 2 , appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincaré map, in action-angle coordinates.

Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators.

Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.

Entanglement in interacting quenched two-body coupled oscillator system

In this work, we explore the effects of a quantum quench on the entanglement measures of a two-body coupled oscillator system having quartic interaction. We use the invariant operator method, under a perturbative framework, for computing the ground state of this system. We give the analytical expressions for the total and reduced density matrix of the system having non-Gaussian, quartic interaction terms. Using this reduced density matrix, we show the analytical calculation of two entanglement measures viz., Von Neumann entanglement entropy using replica trick and Renyi entanglement entropy. Further, we give a numerical estimate of these entanglement measures with respect to the dimensionless parameter $(t/\delta t$) and show its behaviour in the three regimes, i.e; late time behaviour, around the quench point and the early time behaviour. We comment on the variation of these entanglement measures for different orders of coupling strength. The variation of Renyi entropy of different orders has also been discussed.

Learning Coupled Oscillators System with Reservoir Computing

In this paper, we reconstruct the dynamic behavior of the ring-coupled Lorenz oscillators system by reservoir computing. Although the reconstruction of various complex chaotic attractors has been well studied by using various neural networks, little attention has been paid to whether the spatio-temporal structure of some special attractors can be maintained in long-term prediction. Reservoir computing has been shown to be effective for model-free prediction, so we want to investigate whether reservoir computing can restore the rotational symmetry of the original ring-coupled Lorenz system. We find that although the state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated spatio-temporal structure is maintained in the process of reconstruction. Specifically, we show that the rotational symmetric structure of periodic rotating waves, quasi-periodic torus, and chaotic rotating waves is well maintained.

Stability of a parametrically driven, coupled oscillator system: An auxiliary function method approach

Coupled, parametric oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the long-term statistics of the oscillator system’s trajectories and stability. We present a novel, robust, and computationally efficient method, which has come to be known as an auxiliary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxiliary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability region across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability region and the approximate stability region can occur at physically relevant parameter values in regions away from parametric resonance. As an alternative to asymptotic methods, we show that the auxiliary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.

Spacing statistics of energy spectra: random matrices, black hole thermalization, and echoes

Recent advances in AdS/CFT holography have suggested that the near-horizon dynamics of black holes can be described by random matrix systems. We study how the energy spectrum of a system with a generic random Hamiltonian matrix affects its early and late time thermalization behaviour using the spectral form factor (which captures the time-dependence of two-point correlation functions). We introduce a simple statistical framework for generating random spectra in terms of the nearest neighbor spacing statistics of energy eigenvalues, enabling us to compute the averaged spectral form factor in a closed form. This helps to easily illustrate how the spectral form factor changes with different choices of nearest neighbor statistics ranging from the Poisson to Wigner surmise statistics. We suggest that it is possible to have late time oscillations in random matrix models involving β-ensembles (generalizing classical Gaussian ensembles). We also study the form factor of randomly coupled oscillator systems and show that at weak coupling, such systems exhibit regular decaying oscillations in the spectral form factor making them interesting toy models for gravitational wave echoes. We speculate on the holographic interpretation of a system of coupled oscillators, and suggest that they describe the thermalization behaviour of a black hole geometry with a membrane that cuts off the geometry at the stretched horizon.

Analytical expressions for real and complex Fano parameters in a simple classical harmonic oscillator system

We analytically studied the Fano resonance in a simple coupled oscillator system. We demonstrate directly from the equation of motion that the resonance profile observed in this system is generally described by the Fano formula with a complex Fano parameter. The analytical expressions are derived for resonance frequency, resonance width, and Fano parameter, moreover conditions under which the Fano parameter becomes a real number are also examined. These expressions, derived for the simple system, are expected to be helpful for considering various other physical systems because the Fano resonance is a general wave phenomenon.

Synchronization, symmetry and rotating periodic solutions in oscillators with Huygens’ coupling

Synchronization is widespread in complex systems made by multiple individuals. And rotating waves are usually used to describe the synchronization in coupled oscillators systems. Many systems are observed to produce patterns of rotating waves, but it is difficult to predict the type of them or understand the conditions under which they form. Here we present the concept of the rotating-periodic solution and develop new bifurcation theory about rotating-periodic solutions to show the mechanisms for the existence of various rotating waves.We use a Huygens’ coupling model to study this topic. All kinds of rotating waves like in-phase, anti-phase, periodic, cluster synchronous are different types of rotating-periodic solutions with different rotating matrices. In the symmetric Huygens model, the rotating matrices Q satisfying the system rotating invariance just form a special symmetry group. By using the conjugate classes of symmetry groups and the diagonalization method, we can obtain all kinds of rotating waves of finite identical oscillator systems. We calculate all possible rotating waves in three and four identical oscillator systems, and get a general result that the phase difference of the oscillators in a rotating wave can only be kπn (n is the number of the oscillators, 0≤k≤n, k is even). Furthermore, in order to obtain the existence of rotating waves, we establish a new rotating periodic solution Hopf bifurcation theory. This particular bifurcation theory can be used to replace Hopf bifurcation, double Hopf bifurcation and other more complex periodic solution bifurcation in a unified way to study the rotating waves.

New criteria of almost sure exponential stability and instability of nonlinear stochastic systems with a generalization to stochastic coupled systems

This paper focuses on the almost sure exponential stability and instability problems for nonlinear stochastic systems. In contrast with previous works, combining with Lyapunov’s second method and inequalities techniques, two classes of less conservative criteria are obtained, depending on piecewise continuous scalar functions (PCSFs). It is highly desirable that the PCSF can not only replace the constant coefficient of the upper bound estimation which belongs to the diffusion operator of a Lyapunov function in available literature but also even be unbounded. Especially, when PCSFs are assumed to be periodic, our theoretical results can be transformed into the case of almost sure exponential stability and instability of nonlinear systems with intermittent noises, and corresponding criteria are derived. Furthermore, the sufficient criteria we present are popularized to stochastic coupled systems taking advantage of graph theory and Lyapunov method. As an application of theoretical results, we put forward an exploration to the synchronization analysis of a class of single-link robot arms system via intermittent noise control. Finally, several numerical examples consisting of a time-varying stochastic system and a stochastic coupled oscillators system as well as single-link robot arms systems are provided to illustrate the feasibility of the stated theoretical results.

2Pb1-3 Resonance profiles and complex Fano parameters in a weakly coupled oscillator system with degenerate eigenfrequencies

2Pb1-3 Resonance profiles and complex Fano parameters in a weakly coupled oscillator system with degenerate eigenfrequencies

On the Resonances of Coupled Qubit–Cavity Systems

We consider a system consisting of a qubit and a microwave transmission line that are coupled by a capacitor which, in turn, is modulated sinusoidally. The Unruh effect is the simultaneous production from vacuum of a pair of photons, one in the qubit and the other in the cavity. The dynamical Casimir effect is the production from vacuum of a pair of photons in the cavity. We analyze this qubit–cavity system and show that the system can be viewed as a pair of coupled quantum-mechanical oscillators and that both the Unruh effect and the dynamical Casimir effect are resonances of this coupled oscillator system. For the case where the cavity supports two propagating modes, in addition to the Unruh and dynamical Casimir effect at each of the supported modes, we predict a “paired Casimir effect,” where one photon is emitted in the cavity in each of two allowed modes, at the appropriate driving frequency. We also calculate analytical approximations to the driving frequencies for all three effects.

Interfacial anisotropic exciton-polariton manifolds in ReS2

Light–matter coupling in van der Waal’s materials holds significant promise in realizing bosonic condensation and superfluidity. The underlying semiconductor’s crystal asymmetry, if any, can be utilized to form anisotropic half-light half-matter quasiparticles. We demonstrate generation of such highly anisotropic exciton-polaritons at the interface of a biaxial layered semiconductor, stacked on top of a distributed Bragg reflector. The spatially confined photonic mode in this geometry couples with polarized excitons and their Rydberg states, creating a system of highly anisotropic polariton manifolds, displaying Rabi splitting of up to 68 meV. Rotation of the incident beam polarization is used to tune coupling strength and smoothly switch regimes from weak to strong coupling, while also enabling transition from one three-body coupled oscillator system to another. Light–matter coupling is further tunable by varying the number of weakly coupled optically active layers. Our work provides a versatile method of engineering devices for applications in polarization-controlled polaritonics and optoelectronics.

Instability of Meissner Differential Equation and Its Relation with Photon Excitations and Entanglement in a System of Coupled Quantum Oscillators

In this work, we investigate the Schrödinger dynamics of photon excitation numbers and entanglement in a system composed by two non-resonant time-dependent coupled oscillators. By considering π periodically pumped parameters (oscillator frequencies and coupling) and using suitable transformations, we show that the quantum dynamics can be determined by two classical Meissner oscillators. We then study analytically the stability of these differential equations and the dynamics of photon excitations and entanglement in the quantum system numerically. Our analysis shows two interesting results, which can be summarized as follows: (i) Classical instability of classical analog of quantum oscillators and photon excitation numbers (expectations Nj) are strongly correlated, and (ii) photon excitations and entanglement are connected to each other. These results can be used to shed light on the link between quantum systems and their classical counterparts and provide a nice complement to the existing works studying the dynamics of coupled quantum oscillators.

Insights Into the Dynamics of Coupled VO2 Oscillators for ONNs

The collective behavior of many coupled oscillator systems is currently being explored for the implementation of different non-conventional computing paradigms. In particular, VO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> based nano-oscillators have been proposed to implement oscillatory neural networks that can serve as associative memories, useful in pattern recognition applications. Although the dynamics of a pair of coupled oscillators have already been extensively analyzed, in this brief, the topic is addressed more practically. Firstly, for the application mentioned above, each oscillator needs to be initialized in a given phase to represent the input pattern. We demonstrate the impact of this initialization mechanism on the final phase relationship of the oscillators. Secondly, such oscillatory networks are based on frequency synchronization, in which the impact of variability is critical. We carried out a comprehensive mathematical analysis of a pair of coupled oscillators taking into account both issues, which is a first step towards the design of the oscillatory neural networks for associative memory applications.

The lowest frequency modes of the mandolin modeled as a coupled oscillator system

The mandolin is an acoustically unique musical instrument with elements similar to the guitar and violin. Inspired by a classic guitar study by Christensen and Vistisen, we tested if the mandolin’s low-frequency response could be modeled as a simple 3-mass coupled oscillator system. We measured the response of the front plate, back plate, and air within thecavity from sinusoidal forcing of the front plate using microphones, accelerometers, and a force probe. We calculated the mobility (i.e., 1/impedance = v/F) and compared this to a theoretical model. To determine how the collective resonances are affected by the individual oscillators in the system, we added small masses to the front and back plates, and “collared” the f-holes to add mass to the moving air, thus lowering the resonances of the corresponding oscillator. We found clear evidence of coupling: by changing any one element, the resulting three resonant frequencies of the coupled system were affected. We also found the expected phase relations between the individual oscillators for each of the three resonances of the coupled system. From a detailed analysis, we found that a 3-mass coupled oscillator model can reasonably approximate the low-frequency behavior of the mandolin.

Multiband transparency effect induced by toroidal excitation in a strongly coupled planar terahertz metamaterial

The multiband transparency effect in terahertz (THz) domain has intrigued the scientific community due to its significance in developing THz multiband devices. In this article, we have proposed a planar metamaterial geometry comprised of a toroidal split ring resonator (TSRR) flanked by two asymmetric C resonators. The proposed geometry results in multi-band transparency windows in the THz region via strong near field coupling of the toroidal excitation with the dipolar C-resonators of the meta molecule. The geometry displays dominant toroidal excitation as demonstrated by a multipolar analysis of scattered radiation. High Q factor resonances of the metamaterial configuration is reported which can find significance in sensing applications. We report the frequency modulation of transparency windows by changing the separation between TSRR and the C resonators. The numerically simulated findings have been interpreted and validated using an equivalent theoretical model based upon three coupled oscillators system. Such modeling of toroidal resonances may be utilized in future studies on toroidal excitation based EIT responses in metamaterials. Our study has the potential to impact the development of terahertz photonic components useful in building next generation devices.

Entanglement of nitrogen-vacancy-center ensembles with initial squeezing

In this paper, we investigate entanglement of an experimental system of two nitrogen-vacancy-center ensembles which are initially squeezed under the one-axis twisting Hamiltonian. We take into account three scenarios in which initial squeezing and entanglement are mediated by phonons or photons: (a) the phonon-squeezed photon-entangled scenario, (b) the phonon-squeezed phonon-entangled scenario, and (c) the photon-squeezed photon-entangled scenario. For our investigation, we employ the Tavis-Cummings model, which includes dissipative decoherence of the collective spin ensemble, and analyze the system both for a relatively small number of spins and in the limit of a large number of spins using the approach of a quantum master equation. Although evidence in the literature on idealized coupled oscillator systems and coupled quantum kicked tops suggests that initial squeezing can enhance entanglement, we find that, in the realistic system studied in this paper, initial squeezing can improve entanglement overall when the field mode interacts in a particular manner with the two spin ensembles. Our analysis using the Holstein-Primakoff transformation and Wigner characteristic function in the rotating frame of reference shows that the entanglement enhancement is a subtle consequence of the way in which the dissipative decoherence rotates the state of the collective spin ensemble such that enhancement depends on the time-evolved rotated states between the presence and absence of initial squeezing.