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.

Enhanced Synchronization of Intercellular Calcium Oscillations by Noise Contaminated Signals

We consider the dynamics of locally coupled calcium oscillation systems,each cell is subjected to extracel-lular contaminated signal,which contains common sub-threshold signal and independent Gaussian noise.It is found thatintermediate noise can enhance synchronized oscillations of calcium ions,where the frequency of noise-induced oscilla-tions is matched with the one of sub-threshold external signal.We show that synchronization is enhanced as a result ofthe entrainment of external signal Furthermore,the effect of coupling strength is considered.We find above-mentionedphenomenon exists only when coupling strength is very small.Our findings may exhibit that noise can enhance thedetection of feeble external signal through the mechanism of synchronization of intercellular calcium ions.

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.

Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp（4,R） Lie Algebra

Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4, R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4, R) U(2).

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.

Application of Coordinate-Momentum Intermediate Representation in Solving Eigenstate Problem of a Coupled Oscillator System

We study the eigenstate problem of a kind of coupled oscillators in the new quantum mechanical representation ｜q,μ,υ〉, which is defined as the eigenvector of the operator （μQ ＋ υP）, whereμ and υ are two real parameters. We also use the U operator transformation method to deal with the same problem. We obtain the normally ordered product expressions of U operator and eigenvector. It is shown that the ground state of system Hamiltonian is a squeezed state.

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.

Transition to Amplitude Death in Coupled System with Small Number of Nonlinear Oscillators

In this work, we investigate the amplitude death in coupled system with small number of nonlinear oscillators. We show how the transitions to the partial and the complete amplitude deathes happen. We also show that the partial amplitude death can be found in globally coupled oscillators either.

Synchronization of Spatiotemporal Chaos in Coupled Complex Ginzburg—Landau Oscillators

Synchronization of spatiotemporal distributed system is investigated by considering the model of 1D dif-fusively coupled complex Ginzburg-Landau oscillators. An itinerant approach is suggested to randomly move turbulentsignal injections in the space of spatiotemporal chaos. Our numerical simulations show that perfect turbulence synchro-nization can be achieved with properly selected itinerant time and coupling intensity.

Application of Tripartite Entangled State Representation in Solving the Dynamics of a Kind of Three Coupled Oscillators

We study the eigenstate problem of a kind of three coupled oscillators in anew quantum mechanical representation composed by the spontaneous eigenvectors 【 p,q_2,q_3| forthree operators (p_1 + p_2 + p_3), (x_3 - x_2), and (x_3 - x_1). The eigenvalues and eigenvectors ofthe Hamiltonian are obtained. With the same method, the eigenstate problem of a generalizedthree-coupled oscillator Hamiltonian is studied, which has never been studied before.

Energy Spectrum of a Bipartite Complicated-Coupled-Oscillator System

We present a general discussion on the eigenstate problem of a bipartite complicated-coupled-oscillator system. By use of a generalized intermediate entangled state representation, the eigenvalue and eigenfunction of Hamiltonian are analytically obtained. As its application, we obtain the energy spectrum for two special situations of Hamiltonian.

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.

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.

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.

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.