Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.     

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.     

Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators

We analyze a minimal model of a population of identical oscillators with a nonlinear coupling—a generalization of the popular Kuramoto model. In addition to well-known for the Kuramoto model regimes of full synchrony, full asynchrony, and integrable neutral quasiperiodic states, ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics: self-organized bunch states and self-organized quasiperiodic dynamics. The analysis based on the Watanabe-Strogatz ansatz allows us to describe the self-organized bunch states in any finite ensemble as a set of equilibria, and the self-organized quasiperiodicity as a two-frequency quasiperiodic regime. An analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed.     

Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators

A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded.     

混沌吸引子在两个周期振子耦合下的相同步

在分析了系统稳定的基础上，对非线性混沌吸引子在两个独立外周期振子耦合下的相同步进 行了研究.与一个周期振子耦合的情况不同，两个周期振子对混沌吸引子的耦合具有排他性 和竞争性，相同步在两个亚稳态交替出现，各自同步时间长度由外振子参数决定.确定了周 期外振子参数与同步时间长度的关系并与数值模拟计算结果进行了比较. 关键词： 混沌吸引子 相同步     

Phase clustering in globally coupled photochemical oscillators

We experimentally investigate the formation of clusters in a population of globally coupled photochemical oscillators. The system consists of catalytic micro-particles in Belousov-Zhabotinsky solution and the coupling exploits the excitatory properties of light; an increase in the light intensity leads to excitation (“firing") of an oscillator. As the coupling strength is increased, a transition occurs from incoherence to clustering, whereby the oscillators split into synchronised groups, to complete synchronisation. Multistability is observed between a one-phase cluster (fully synchronised group) and two-phase clusters (two groups with the same frequency but different phases). The results are reproduced in simulations and we demonstrate that the heterogeneity of the population as well as the relaxational nature of the oscillators is important in the observation of clusters. We also examine the exploitation of the phase model for the prediction of clusters in experiments.     

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.     

Phase compactons in chains of dispersively coupled oscillators

We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.     

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also presents multistability shown in the planes of the basin of attractions.     

A bifurcation analysis of two coupled calcium oscillators

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. (c) 2001 American Institute of Physics.     

Phase equation with nonlinear excitation for nonlocally coupled oscillators

Some reaction-diffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is effectively a diffusion equation which is excitation-free. However, more complex forms are possible such as the Nikolaevskii equation and the Kuramoto–Sivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.     

用两耦合非线性振子模型研究量子纠缠和相干性

我们研究两耦合非线性振子模型的量子纠缠和量子相干动力学, 结果表明在适当条件下, 纠缠熵与相干度是正相关的. 我们也讨论纠缠熵和相干度与两振子子相互作用能的联系.     

用两耦合非线性振子模型研究量子纠缠和相干性

我们研究两耦合非线性振子模型的量子纠缠和量子相干动力学,结果表明在适当条件下,纠缠熵与相干度是正相关的.我们也讨论纠缠熵和相干度与两振子相互作用能的联系.     

Phase diagram for the Winfree model of coupled nonlinear oscillators

In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other large populations of biological oscillators. Here we give the first bifurcation analysis of the model, for a tractable special case. The system displays rich collective dynamics as a function of the coupling strength and the spread of natural frequencies. Besides incoherence, frequency locking, and oscillator death, there exist hybrid solutions that combine two or more of these states. We present the phase diagram and derive several of the stability boundaries analytically.     

Phase locking in networks of synaptically coupled McKean relaxation oscillators

We use geometric dynamical systems methods to derive phase equations for networks of weakly connected McKean relaxation oscillators. We derive an explicit formula for the connection function when the oscillators are coupled with chemical synapses modeled as the convolution of some input spike train with an appropriate synaptic kernel. The theory allows the systematic investigation of the way in which a slow recovery variable can interact with synaptic time scales to produce phase-locked solutions in networks of pulse coupled neural relaxation oscillators. The theory is exact in the singular limit that the fast and slow time scales of the neural oscillator become effectively independent. By focusing on a pair of mutually coupled McKean oscillators with alpha function synaptic kernels, we clarify the role that fast and slow synapses of excitatory and inhibitory type can play in producing stable phase-locked rhythms. In particular we show that for fast excitatory synapses there is coexistence of a stable synchronous, a stable anti-synchronous, and one stable asynchronous solution. For slower synapses the anti-synchronous solution can lose stability, whilst for even slower synapses it can regain stability. The case of inhibitory synapses is similar up to a reversal of the stability of solution branches. Using a return-map analysis the case of strong pulsatile coupling is also considered. In this case it is shown that the synchronous solution can co-exist with a continuum of asynchronous states.     

双模耦合谐振子哈密顿量的一般解法

双模耦合谐振子哈密顿量是广义坐标■和广义动量■的一般二次型,通过一个坐标变换可以将其表示为新基底下的标准二次型,经计算得知,新基底之间满足准正则对易关系,从而引入准粒子的产生和湮没算符,这样就消除了耦合项,哈密顿量化简成为双模独立谐振子情形,使问题得到解决.这样的解决方法可以推广到各向异性n模谐振子的耦合体系.     

Vibrational and stochastic resonances in two coupled overdamped anharmonic oscillators

We study the overdamped version of two coupled anharmonic oscillators under the influence of both low- and high-frequency forces respectively and a Gaussian noise term added to one of the two state variables of the system. The dynamics of the system is first studied in the presence of both forces separately without noise. In the presence of only one of the forces, no resonance behaviour is observed, however, hysteresis happens there. Then the influence of the high-frequency force in the presence of a low-frequency, i.e. biharmonic forcing, is studied. Vibrational resonance is found to occur when the amplitude of the high-frequency force is varied. The resonance curve resembles a stochastic resonance-like curve. It is maximum at the value of g at which the orbit lies in one well during one half of the drive cycle of the low-frequency force and in the other for the remaining half cycle. Vibrational resonance is characterized using the response amplitude and mean residence time. We show the occurrence of stochastic resonance behaviour in the overdamped system by replacing the high-frequency force by Gaussian noise. Similarities and differences between both types of resonance are presented.     

Synchronized oscillations in a system of two coupled van-der-pol-duffing oscillators

The results of analysis of the periodic solutions obtained within the framework of complete and truncated equations for a system of identical Van-der-Pol-Duffing oscillators with nonlinear coupling are compared. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1531–1536, December, 1998.     

不同质量和频率的耦合谐振子体系哈密顿量的对角化

讨论了耦合项为λ１Ｘ１Ｘ２和λ２ｐ１ｐ２的两个不同质量和频率的耦合谐振子哈密顿量的对角化。     

Two quantum oscillators coupled with a planar radio frequency ion trap

In this scheme,two quantum oscillators in a planar radio frequency ion trap are coupled by the trap electrodes.The ions motional states encode the quantum bits (qubits),and a swap gate could be achieved.Under different conditions of the experiments,the intensity of the coupling between two quantum oscillators and the dissipation of the system are calculated.We compute fidelities for a quantum swap gate and discuss experimental issues.