Phase and frequency shifts of two nonlinearly coupled oscillators

We analyze two nonlinearly phase coupled oscillators with eigenfrequencies ω₁and ω₂, where n\gw₁=m\gw₂+\gp, with integern andm. For \gh=0 there are up to four stable synchronized states which differ from each other only by the difference of the oscillators\rs phases. The number of different synchronized states depends on the coupling constants. If \gh does not vanish phase shifts and frequency shifts may occur givig rise to stable synchronized states which also differ from each other due to the frequencies. By means of the center manifold theorem we calculate these shifts explicitely. Different coupling constants are investigated: symmetrical, homogenously asymmetrical and arbitrary coupling constants. Our results point out the decisive influence of the symmetry of the coupling constants upon the frequency and phase shifts. Moreover the local stability of the unperturbed synchronized state (i.e. for \gh=0) determines the magnitude of the frequency and phase shifts.     

Synchronization of oscillators coupled through an environment

We study synchronization of oscillators that are indirectly coupled through their interaction with an environment. We give criteria for the stability or instability of a synchronized oscillation. Using these criteria we investigate synchronization of systems of oscillators which are weakly coupled, in the sense that the influence of the oscillators on the environment is weak. We prove that arbitrarily weak coupling will synchronize the oscillators, provided that this coupling is of the ‘right’ sign. We illustrate our general results by applications to a model of coupled GnRH neuron oscillators proposed by Khadra and Li [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74-83.], and to indirectly weakly-coupled λ-ω oscillators.     

Transitions to synchrony in coupled bursting neurons

Certain cells in the brain, for example, thalamic neurons during sleep, show spike-burst activity. We study such spike-burst neural activity and the transitions to a synchronized state using a model of coupled bursting neurons. In an electrically coupled network, we show that the increase of coupling strength increases incoherence first and then induces two different transitions to synchronized states, one associated with bursts and the other with spikes. These sequential transitions to synchronized states are determined by the zero crossings of the maximum transverse Lyapunov exponents. These results suggest that synchronization of spike-burst activity is a multi-time-scale phenomenon and burst synchrony is a precursor to spike synchrony.     

Glassy synchronization in a population of coupled oscillators

A phase model for a population of oscillators with random excitatory and inhibitory mean-field coupling and subject to external white noise random forces is proposed and studied. In the thermodynamic limit different stable phases for the oscillator population may be found: (i) an incoherent state where all possible values of an oscillator phase are equally probable, (ii) a synchronized state where the population has a nonzero collective phase; (iii) a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder; and (iv) a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder. These predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by Brownian simulation.     

Transport properties of coupled Brownian particles confined to a microchannel in an asymmetric periodic potential

On the basis of the transport features and experimental phenomena observed in studies of molecular motors, we investigated an overdamped Brownian motion of two coupled particles with an asymmetric periodic potential in a two-dimensional microchannel theoretically and numerically, to reveal the dynamical mechanism of cooperative transport of particles with two heads, where the interactions between two particles are taken into consideration. Moreover, while moving in a confined structure, Brownian particles also could exhibit peculiar kinetic behavior. The dependence of directed current on various parameters is systematically studied. Our results indicate that the direction of motion can be reversed by modulating the coupling strength, free length, and microchannel width. In addition, we have achieved the conditions of forward motion in this study. That is, when the interparticle average horizontal interval Δx > 0.25L, where L is the spatial period of the external potential, the forward motion of coupled Brownian particles effected by the synchronized noise and confined to a microchannel can be generated in the strong-coupling case.     

Diffusion-model analysis of effective CIDEP distance in solvent-separated radical-ion pair

The radical pair mechanism (RPM) of chemically induced dynamic electron polarization (CIDEP) is theoretically analyzed to determine what intermolecular separations (r _eff) effectively contribute to the CIDEP generated from diffusive, separated radical-ion pairs (RIP) in terms of the chargetransfer interaction in the singlet-triplet energy splitting (J) by taking into account the distance-dependent electronic coupling and reorganization energy. The diffusion-model analysis reveals that the hyperfine-dependent RPM polarization (P _RPM) phase is varied with the driving force (?ΔG _CR) for the charge-recombination (CR) process and that the boundary ?ΔG _CR between the opposite phases coincides well with the total reorganization energy around the diffusible separation distance,r _eff=1.2 nm, between the ion radicals. For the first time, ther _eff is well described by the exponent parameter (β) in the distance-dependent electronic coupling, suggesting that the RPM CIDEP detection can be applied to characterize the electronic coupling in individual solvent-separated RIP systems. It has been concluded that, in contrast to the spin exchange interaction of the neutral radical pairs, the characteristic long-range charge-transfer interaction enables us to utilize the simple diffusion-model analysis to successfully evaluate ther _eff and theP _RPM in homogeneous liquid polar solvents.     

Collective behavior in coupled chaotic map lattices with random perturbations

Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a “thermal bath”, where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x→4x(1−x).     

Continuation of normal modes in finite NLS lattices

We study the continuation of breather solutions of the discrete NLS equation as the intersite coupling parameter is varied. Considering the case of a finite one-dimensional lattice of N sites, we show the existence of N branches of breathers that persist for arbitrary coupling, thus connecting normal modes of the linear system to breathers of the uncoupled, anticontinuous limit system. The proof is based on global bifurcation theory, applied to the continuation from the weakly nonlinear regime. As the coupling parameter varies these solutions generally change their stability, and we detect parameter regions where trajectories starting near unstable breathers appear to reach equipartition of power.     

On the arithmetic of phase locking: Coupled neurons as a lattice on R2

Using methods from the geometry of numbers, we derive an explicit, global solution for the phase-locking behavior of a simple integrate-and-fire model of coupled neurons. The solution gives the ratios of phase locking (rotation numbers) attained as functions of the parameters of natural frequency and bidirectional coupling. The ordering of the ratios is related to Farey-type series and to simple continued fractions. A transition between two ratios, say $\text{a}\text{b}$ to $\text{c}\text{d}$, is possible if, and only if, ad?bc=±1. Empirically, similar ordering is evident in published data from various neuron analogues. We compare and contrast the present results with those from models based on Caianiello's equation and on more general mappings on the torus.     

Global chaos synchronization of coupled parametrically excited pendula

In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.     

Enhancement of neural synchrony by time delay

In a network of neuronal oscillators with time-delayed coupling, we uncover a phenomenon of enhancement of neural synchrony by time delay: a stable synchronized state exists at low coupling strengths for significant time delays. By formulating a master stability equation for time-delayed networks of Hindmarsh-Rose neurons, we show that there is always an extended region of stable synchronous activity corresponding to low coupling strengths. Such synchrony could be achieved in the undelayed system only by much higher coupling strengths. This phenomenon of enhanced neural synchrony by delay has important implications, in particular, in understanding synchronization of distant neurons and information processing in the brain.     

Observation of double-Pomeron exchange in alpha-alpha collisions at the CERN intersecting storage rings

We observed the process αα→ααX in which the α′s were emitted uncorrelated in the forward direction and the charged component of the clusterX was confined to a limited portion (|η|?2) of the central region. We identified such reactions as being due to double-Pomeron exchange, for which we found a cross-section of (720±140)μb. The raw mean charged multiplicity of the clusterX was found to be 6.76±0.07 with a dispersionD=4.8. The measurements were performed at the CERN ISR at a centre-of-mass energy of \(\sqrt s = 126\) GeV. Similarities are drawn between double-Pomeron exchange in αα and inpp collisions.     

Noise-driven synchronization of a FitzHugh-Nagumo ring with phase-repulsive coupling: A perspective from the system’s nonequilibrium potential

We study a one-dimensional array of N autonomous units with excitable FitzHugh-Nagumo dynamics coupled in phase-repulsive way to form a ring, and submitted to a common subthreshold harmonic signal and independent Gaussian white noises with a common intensity η. By varying η, two macroscopic regimes are observed. For some value of noise intensity, a transition from the rest state to an activated one-with almost half of the neurons excited forming an “...-activated-inhibited-activated-... ” structure along the ring-takes place. For larger values of η, the inverse transition is also observed, and both states alternate in a synchronized way with the signal. Moreover, measures of activation and coherent behavior become maximal for intermediate values of η. The origin of these collective effects is explained in terms of the system’s nonequilibrium potential. In particular, the levels of noise for activation and synchronization are theoretically estimated.     

On the chiral covariant approach to ρρ scattering

We examine in detail a recent work(D. Gülmez, U. G. Mei?ner and J. A. Oller, Eur. Phys. J. C,77: 460(2017)), where improvements to make ρρ scattering relativistically covariant are made. The paper has the remarkable conclusion that the J =2 state disappears with a potential which is much more attractive than for J =0,where a bound state is found. We trace this abnormal conclusion to the fact that an "on-shell" factorization of the potential is done in a region where this potential is singular and develops a large discontinuous and unphysical imaginary part. A method is developed, evaluating the loops with full ρ propagators, and we show that they do not develop singularities and do not have an imaginary part below threshold. With this result for the loops we define an effective potential, which when used with the Bethe-Salpeter equation provides a state with J =2 around the energy of the f_2(1270). In addition, the coupling of the state to ρρ is evaluated and we find that this coupling and the T matrix around the energy of the bound state are remarkably similar to those obtained with a drastic approximation used previously, in which the q~2 terms of the propagators of the exchanged ρ mesons are dropped, once the cut-off in the ρρ loop function is tuned to reproduce the bound state at the same energy.     

Generalized Turing patterns and their selective realization in spatiotemporal systems

We consider the pattern formation problem in coupled identical systems after the global synchronized state becomes unstable. Based on analytical results relating the coupling strengths and the instability of each spatial mode (pattern) we show that these spatial patterns can be selectively realized by varying the coupling strengths along different paths in the parameter space. Furthermore, we discuss the important role of the synchronized state (fixed point versus chaotic attractor) in modulating the temporal dynamics of the spatial patterns.     

Bipolarons in the intermediate coupling method

The properties of polarons and bipolarons are studied by the intermediate coupling method, which takes electron-lattice coupling into account. A Coulomb correlation in the electron motion is included. When the Fröhlich coupling constant is reduced, at α*=5.7 a bipolaron decays suddenly from a self-trapped state into two delocalized polarons. A phase diagram is constructed for the region where the stable bipolaron exists. These results are compared with those obtained by integration along trajectories.     

Populations of coupled electrochemical oscillators

Experiments were carried out on arrays of chaotic electrochemical oscillators to which global coupling, periodic forcing, and feedback were applied. The global coupling converts a very weakly coupled set of chaotic oscillators to a synchronized state with sufficiently large values of coupling strength; at intermediate values both intermittent and stable chaotic cluster states occur. Cluster formation and synchronization were also obtained by applying feedback and forcing to a moderately coupled base state. The three cases differ, however, in other details. The feedback and forcing also produce periodic cluster states and more than two clusters. Configurations of two (chaotic) clusters and two, three, or four (periodic) clusters were observed. (c) 2002 American Institute of Physics.     

Mean field approximation for noisy delay coupled excitable neurons

Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit time-delays, modelled by N?1 stochastic delay-differential equations is derived. The resulting approximation contains only two deterministic delay-differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system.     

Josephson current in superconductor/ferromagnet/superconductor junctions

By using the Bogoliubov–de Gennes equation and the Nambu spinor Greens function approach, we have theoretically studied the dc Josephson current and the coupling phase state of superconductor/ferromagnet/superconductor (SC/FM/SC) junctions, where the FM is of weak ferromagnetism. From the behavior of the temperature-dependent dc Josephson current (I_c), we confirm that such SC/FM/SC junction may change from 0-phase to π-phase state with increasing the temperature (T), for particular parameters of the thickness and the strength of ferromagnetism of the FM interlayer. We attribute such changement to an extra phase difference between the two SCs. The results are qualitatively consistent with an experiment [Phys. Rev. Lett. 86 (2001) 2427], which shows a sharp cusp structure on the I_c−T curves of Nb/Cu_0.48Ni_0.52/Nb junction for specific thickness of the Cu_0.48Ni_0.52, indicating the junction changes from 0-phase state at high temperatures to π-phase state at low temperatures.     

Multistability of clustered states in a globally inhibitory network

We study a network of m identical excitatory cells projecting excitatory synaptic connections onto a single inhibitory interneuron, which is reciprocally coupled to all excitatory cells through inhibitory synapses possessing short-term synaptic depression. We find that such a network with global inhibition possesses multiple stable activity patterns with distinct periods, characterized by the clustering of the excitatory cells into synchronized sub-populations. We prove the existence and stability of n-cluster solutions in a m-cell network. Using methods of geometric singular perturbation theory, we show that any n-cluster solution must satisfy a set of consistency conditions that can be geometrically derived. We then characterize the basin of attraction of each solution. Although frequency dependent depression is not necessary for multistability, we discuss how it plays a key role in determining network behavior. We find a functional relationship between the level of synaptic depression, the number of clusters and the interspike interval between neurons. This relationship is much less pronounced in the absence of depression. Implications for temporal coding and memory storage are discussed.