Stability and multistability of synchronization in networks of coupled phase oscillators

Coupled phase oscillators usually achieve synchronization as the coupling strength among oscillators is increased beyond a critical value. The stability of synchronous state remains an open issue. In this paper, we study the stability of the synchronous state in coupled phase oscillators. It is found that numerical integration of differential equations of coupled phase oscillators with a finite time step may induce desynchronization at strong couplings. The mechanism behind this instability is that numerical accumulated errors in simulations may trigger the loss of stability of the synchronous state.Desynchronization critical couplings are found to increase and diverge as a power law with decreasing the integral time step. Theoretical analysis supports the local stability of the synchronized state. Globally the emergence of synchronous state depends on the initial conditions. Other metastable ordered states such as twisted states can coexist with the synchronous mode. These twisted states keep locally stable on a sparse network but lose their stability when the network becomes dense.

Chimera states of phase oscillator populations with nonlocal higher-order couplings

The chimera states underlying many realistic dynamical processes have attracted ample attention in the area of dynamical systems.Here, we generalize the Kuramoto model with nonlocal coupling incorporating higher-order interactions encoded with simplicial complexes.Previous works have shown that higher-order interactions promote coherent states.However, we uncover the fact that the introduced higher-order couplings can significantly enhance the emergence of the incoherent state.Remarkably, we identify that the chimera states arise as a result of multi-attractors in dynamic states.Importantly, we review that the increasing higher-order interactions can significantly shape the emergent probability of chimera states.All the observed results can be well described in terms of the dimension reduction method.This study is a step forward in highlighting the importance of nonlocal higher-order couplings, which might provide control strategies for the occurrence of spatial-temporal patterns in networked systems.

The construction of conserved quantities for linearly coupled oscillators and study of symmetries about the conserved quantities

In this paper, the conserved quantities are constructed using two methods. The first method is by making an ansatz of the conserved quantity and then using the definition of Poisson bracket to obtain the coefficients in the ansatz. The main procedure for the second method is given as follows. Firstly, the coupled terms in Lagrangian are eliminated by changing the coordinate scales and rotating the coordinate axes, secondly, the conserved quantities are obtain in new coordinate directly, and at last, the conserved quantities are expressed in the original coordinates by using the inverse transform of the coordinates. The Noether symmetry and Lie symmetry of the infinitesimal transformations about the conserved quantities are also studied in this paper.

Asymptotic solving method for sea-air coupled oscillator ENSO model

The ENSO is an interannual phenomenon involved in the tropical Pacific ocean-atmosphere interaction. In this article, we create an asymptotic solving method for the nonlinear system of the ENSO model. The asymptotic solution is obtained. And then we can furnish weather forecasts theoretically and other behaviors and rules for the atmosphere- ocean oscillator of the ENSO.

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.

Sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays

In this letter, a distributed protocol for sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays is proposed, and a brief procedure of convergence analysis for such algorithm over undirected connected graphs is provided. Furthermore, a simple yet generic criterion is also presented to guarantee synchronized oscillatory motions in coupled harmonic oscillators. Subsequently, the simulation results are worked out to demonstrate the efficiency and feasibility of the theoretical results.

Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation

We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states（QSs） and the repetitive spiking states（SPs） may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.

The Ott-Antonsen Ansatz in Globally Coupled Phase Oscillators

The Ott-Antonsen ansatz provides a powerful tool in investigating synchronization among coupled phase oscil- lators. However, previous works using the ansatz only focused on the evolution of the order parameter and the information on desynchronized oscillators is less discussed. In this work, we show that the Ott-Antonsen ansatz can also be applied to investigate the desynchronous dynamics in coupled phase oscillators. Studying the original Kuramoto model and two of its variants, we find that the dynamics of α（ω）, the coefficient in the Fourier series of the probability density, can give most of the information on the synchronization, for example, the threshold of natural frequency delimiting the oscillators synchronized and desychronized by the mean field, the formulation of the effective frequency ωe （ω） of desynchronous oscillators, and the structure of the graph ωe （ω）.

Collective stochastic resonance behaviors of two coupled harmonic oscillators driven by dichotomous fluctuating frequency

The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain(OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.

Coupled Two-Dimensional Atomic Oscillation in an Anharmonic Trap

In atomic dynamics, oscillation Mong different axes can be studied separately in the harmonic trap. When the trap is not harmonic, motion in different directions may couple together. In this work, we observe a two- dimensional oscillation by exciting atoms in one direction, where the atoms are transferred to an anharmonic region. Theoretical calculations are coincident to the experimental results. These oscillations in two dimensions not only can be used to measure trap parameters but also have potential applications in atomic interferometry and precise measurements.

Numerical analysis of motional mode coupling of sympathetically cooled two-ion crystals

A two-ion pair in a linear Paul trap is extensively used in the research of the simplest quantum-logic system;however,there are few quantitative and comprehensive studies on the motional mode coupling of two-ion systems yet.This study proposes a method to investigate the motional mode coupling of sympathetically cooled two-ion crystals by quantifying three-dimensional(3 D)secular spectra of trapped ions using molecular dynamics simulations.The 3 D resonance peaks of the^(40)Ca^(+)–^(27)Al^(+)pair obtained by using this method were in good agreement with the 3 D in-and out-of-phase modes predicted by the mode coupling theory for two ions in equilibrium and the frequency matching errors were lower than 2%.The obtained and predicted amplitudes of these modes were also qualitatively similar.It was observed that the strength of the sympathetic interaction of the^(40)Ca^(+)–^(27)Al^(+)pair was primarily determined by its axial in-phase coupling.In addition,the frequencies and amplitudes of the ion pair's resonance modes(in all dimensions)were sensitive to the relative masses of the ion pair,and a decrease in the mass mismatch enhanced the sympathetic cooling rates.The sympathetic interactions of the^(40)Ca^(+)–^(27)Al^(+)pair were slightly weaker than those of the^(24)Mg^(+)–^(27)Al^(+)pair,but significantly stronger than those of^(9)Be^(+)–^(27)Al^(+).However,the Doppler cooling limit temperature of^(40)Ca^(+)is comparable to that of^(9)Be^(+)but lower than approximately half of that of^(24)Mg^(+).Furthermore,laser cooling systems for^(40)Ca^(+)are more reliable than those for^(24)Mg^(+)and^(9)Be^(+).Therefore,^(40)Ca^(+)is probably the best laser-cooled ion for sympathetic cooling and quantum-logic operations of^(27)Al^(+)and has particularly more notable comprehensive advantages in the development of high reliability,compact,and transportable^(27)Al^(+)optical clocks.This methodology may be extended to multi-ion systems,and it will greatly aid efforts to control the dynamic behaviors of sympathetic cooling as well as the development of low-heating-rate quantum logic clocks.

New representation of the multimode phase shifting operator and its application

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the mnltimode coordinate-momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the rnultimode phase shifting operator.

Investigation of a mutual interaction force at different pressure amplitudes in sulfuric acid

This paper investiages the secondary Bjerknes force for two oscillating bubbles in various pressure amplitudes in a concentration of 95% sulfuric acid. The equilibrium radii of the bubbles are assumed to be smaller than 10 μm at a frequency of 37 kHz in various strong driving acoustical fields around 2.0 bars （1 bar=10^5 Pa）. The secondary Bjerknes force is investigated in uncoupled and coupled states between the bubbles, with regard to the quasi-adiabatic model for the bubble interior. It finds that the value of the secondary Bjerknes force depends on the driven pressure of sulfuric acid and its amount would be increased by liquid pressure amplitude enhancement. The results show that the repulsion area of the interaction force would be increased by increasing the driven pressure because of nonlinear oscillation of bubbles.

Right on time:measuring Kuramoto model coupling from a survey of wrist-watches

Using a survey of wrist-watch synchronization from a randomly selected group of independent volunteers, we model the system as a Kuramoto-type coupled oscillator network. Based on the phase data both the order parameter and likely size of the coupling are derived and the possibilities for similar research to deduce topology from dynamics are discussed.

Brownian localization:A generalized coupling model yielding a nonergodic Langevin equation description

A minimal system-plus-reservoir model yielding a nonergodic Langevin equation is proposed, which originates from the cubic-spectral density of environmental oscillators and momentum-dependent coupling. This model allows ballistic diffusion and classical localization simultaneously, in which the fluctuation-dissipation relation is still satisfied but the Khinchin theorem is broken. The asymptotical equilibrium for a nonergodic system requires the initial thermal equilibrium, however, when the system starts from nonthermal conditions, it does not approach the equilibration even though a nonlinear potential is used to bound the particle, this can be confirmed by the zerotb law of thermodynamics. In the dynamics of Brownian localization, due to the memory damping function inducing a constant term, our results show that the stationary distribution of the system depends on its initial preparation of coordinate rather than momentum. The coupled oscillator chain with a fixed end boundary acts as a heat bath, which has long been used in studies of collinear atom/solid-surface scattering and lattice vibration, we investigate this problem from the viewpoint of nonergodicity.

CONVERGENCE OF ATTRACTORS

The system of coupled oscillators and its time-discretization (with constantstepsize h) are considered in this paper. Under some conditions, it is showed that thediscrete systems have one-dimensional global attroctors lh converging to I which is theglobal attractor of continuous system.

Relaxation dynamics of Kuramoto model with heterogeneous coupling

The Landau damping which reveals the characteristic of relaxation dynamics for an equilibrium state is a universal concept in the area of complex system. In this paper, we study the Landau damping in the phase oscillator system by considering two types of coupling heterogeneity in the Kuramoto model. We show that the critical coupling strength for phase transition, which can be obtained analytically through the balanced integral equation, has the same formula for both cases. The Landau damping effects are further explained in the framework of Laplace transform, where the order parameters decay to zero in the long time limit.

Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators

Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators,the identification of the symmetries in largesize complex networks remains as a challenge.Additionally,even though the topological symmetries of a complex network are known,it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators.We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and,incorporating the conventional method of eigenvalue-based analysis,investigate the emergence and transition of the cluster synchronization states.We are able to argue and demonstrate that,without a prior knowledge of the network symmetries,the method is able to predict not only all the cluster synchronization states observable in the network,but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition.The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators,including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries.The new framework paves a way to the investigation of synchronization patterns in large-size,general complex networks.

Synchronization of coupled metronomes on two layers

Coupled metronomes serve as a paradigmatic model for exploring the collective behaviors of com- plex dynamical systems, as well as a classical setup for classroom demonstrations of synchronization phenomena. Whereas previous studies of metronome synchronization have been concentrating on symmetric coupling schemes, here we consider the asymmetric case by adopting the scheme of layered metronomes. Specifically, we place two metronomes on each layer, and couple two layers by placing one on top of the other. By varying the initial conditions of the metronomes and adjusting the friction between the two layers, a variety of synchronous patterns are observed in experiment, including the splay synchronization （SS） state, the generalized splay synchronization （GSS） state, the anti-phase synchronization （APS） state, the in-phase delay synchronization （IPDS） state, and the in-phase syn- chronization （IPS） state. In particular, the IPDS state, in which the metronomes on each layer are synchronized in phase but are of a constant phase delay to metronomes on the other layer, is observed for the first time. In addition, a new technique based on audio signals is proposed for pattern detection, which is more convenient and easier to apply than the existing acquisition techniques. Furthermore, a theoretical model is developed to explain the experimental observations, and is employed to explore the dynamical properties of the patterns, including the basin distributions and the pattern transitions. Our study sheds new lights on the collective behaviors of coupled metronomes, and the developed setup can be used in the classroom for demonstration purposes.

Two dimensional coupled oscillators array with rhombus structure and its application in active antenna array

Beam scanning and forming can be achieved by coupled oscillators array without phase shifter. Active antenna array based on coupled oscillators array has the virtue of low cost, high integration, and high efficiency. Traditional two dimensional coupled oscillators array has been arranged on rectangular lattices, and phase difference of adjacent elements is limited to [-90°, 90°]. Therefore, the beam scanning range is limited to [-30°, 30°] from normal for half wavelength element spacing. A new two dimensional coupled oscillators array with rhombus structure is presented. Phase control method and phase error of the array are also provided. Stability of the array is analyzed, and stable condition is given. When this coupled oscillators array with rhombus structure is used in active antenna array, theoretical results show that phase difference of adjacent elements reach the limit of [-180°, 180°] along the horizontal and vertical directions. Therefore, it has wider beam scanning range than that of a rectangular lattice structure.