Minimum Coupling Strength Research Articles

Threshold coupling strength for equilibration between small systems

In this paper we study the thermal equilibration of small bipartite Bose-Hubbard systems, both quantum mechanically and in mean-field approximation. In particular we consider small systems composed of a single-mode "thermometer" coupled to a three-mode "bath", with no additional environment acting on the four-mode system, and test the hypothesis that the thermometer will thermalize if and only if the bath is chaotic. We find that chaos in the bath alone is neither necessary nor sufficient for equilibration in these isolated four-mode systems. The two subsystems can thermalize if the combined system is chaotic even when neither subsystem is chaotic in isolation, and under full quantum dynamics there is a minimum coupling strength between the thermometer and the bath below which the system does not thermalize even if the bath itself is chaotic. We show that the quantum coupling threshold scales like $1/N$ (where $N$ is the total particle number), so that the classical results are obtained in the limit $N\rightarrow \infty$.

Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators.

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

Synchronous firing patterns and transitions in small-world neuronal network

Synchronous firing patterns and transitions of Hindmarsh–Rose small-world neuronal network are studied in this work. With the same external stimuli current, it is shown that the synchronous state of the Newman and Watts (NW) small-world neuronal network is completely determined by its external stimuli current, have nothing about the coupling strength \(C\) as well as the connection probability \(p\); a qualitative explanation is proposed for the opposite changes of the minimum critical coupling strength \(C_{\mathrm{mcc}}\) and the minimum connection probability \(p_{\mathrm{mcp}}\). With two different amplitude stimuli current, only almost complete synchronization can be achieved, and the firing patterns of the synchronized NW small-world neuronal network only distributes the interval with two endpoints, which just correspond to the two modes in a single model neuron with these two kinds of amplitude stimuli current.

Convergence and synchronization in heterogeneous networks of smooth and piecewise smooth systems

This paper presents a framework for the study of convergence in networks where the nodes’ dynamics may be both piecewise smooth and/or nonidentical. Sufficient conditions are derived for global convergence of all node trajectories towards the same bounded region in the synchronization error space. The analysis is based on the use of set-valued Lyapunov functions and bounds are derived on the minimum coupling strength required to make all nodes in the network converge towards each other. We also provide an estimate of the asymptotic bound on the mismatch between the node state trajectories. The analysis is performed both for linear and nonlinear coupling protocols. The theoretical analysis is extensively illustrated and validated via its application to a set of representative numerical examples.

Nearly compensated exchange in the dimer compound callaghaniteCu2Mg2(CO3)(oH)6·2H2O

A combined theoretical and experimental study of the natural Cu${}^{2+}$-mineral callaghanite is presented. Its crystal structure features well separated ${\mathrm{Cu}}_{2}$${\mathrm{(oH)}}_{6}$ structural dimers with weakly bonded carbonate groups and water molecules in between. Susceptibility, field-dependent magnetization and specific-heat measurements reveal a compound with a small spin gap of about 7 K. The observed magnetic properties are well described by a model of isolated antiferromagnetic spin dimers. Possible ferromagnetic interactions between the dimers amount to $\ensuremath{-}1.5$ K, at most. Different flavors of electronic structure calculations have been employed to locate the magnetic dimers in the crystal structure, i.e., to determine whether they coincide with the structural dimers or not. Calculations of the coupling between the structural dimers clearly show that magnetic and structural dimers are the same. For the intradimer coupling, however, the computational results confirmed a coupling strength close to zero but the sign of the coupling could not be determined unambiguously. Based on this finding, we then discuss how the reliability of the numerical methods depends on the characteristics of exchange pathways and on structural features of the compound in general. Eventually, we try to provide a minimum coupling strength that is needed for a reliable computational description.

Adiabatic approximation for three qubits ultrastrongly coupled to a harmonic oscillator

We study the system involving mutual interaction between three qubits and an oscillator within the ultrastrong coupling regime. We apply adiabatic approximation approach to explore two extreme regimes: (i) the oscillator's frequency is far larger than each qubit's frequency and (ii) the qubit's frequency is far larger than the oscillator's frequency, and analyze the energy-level spectrum and the ground-state property of the qubit-oscillator system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum qubit-oscillator coupling strength needed for generating the nonclassical states of the Schr\"{o}dinger-cat type in the oscillator is just one half of that in the Rabi model. We find that the qubit-qubit entanglement in the ground state vanishes if the qubit-oscillator coupling strength is strong enough, for which the entropy of three qubits keeps larger than one. We also observe the phase-transition-like behavior in the regime where the qubit's frequency is far larger than the oscillator's frequency.

Role of network topology in the synchronization of power systems

We study synchronization dynamics in networks of coupled oscillators with bimodal distribution of natural frequencies. This setup can be interpreted as a simple model of frequency synchronization dynamics among generators and loads working in a power network. We derive the minimum coupling strength required to ensure global frequency synchronization. This threshold value can be efficiently found by solving a binary optimization problem, even for large networks. In order to validate our procedure, we compare its results with numerical simulations on a realistic network describing the European interconnected high-voltage electricity system, finding a very good agreement. Our synchronization threshold can be used to test the stability of frequency synchronization to link removals. As the threshold value changes only in very few cases when aplied to the European realistic network, we conclude that network is resilient in this regard. Since the threshold calculation depends on the local connectivity, it can also be used to identify critical network partitions acting as synchronization bottlenecks. In our stability experiments we observe that when a link removal triggers a change in the critical partition, its limits tend to converge to national borders. This phenomenon, which can have important consequences to synchronization dynamics in case of cascading failure, signals the influence of the uncomplete topological integration of national power grids at the European scale.

What controls phase-locking of ENSO to boreal winter in coupled GCMs?

In this study, the phase-locking of El Nino Southern Oscillation (ENSO) in a coupled model with different physical parameter values is investigated. It is found that there is a dramatic change in ENSO phase-locking in response to a slight change in the Tokioka parameter, which is a minimum entrainment rate threshold in the cumulus parameterization. With a smaller Tokioka parameter, the model simulates ENSO peak in the boreal summer season rather than in the winter season as observed. It is revealed that the differences in climatological zonal sea surface temperature (SST) gradient and its associated mean state changes are crucial to determine the phase-locking of ENSO. In the simulations with smaller Tokioka parameter values, climatological zonal SST gradient during the boreal summer is excessively large, because the zonally-asymmetric SST change (i.e., SST increase is relatively smaller over the eastern Pacific) is maximum during the boreal summer when the eastern Pacific SST is the coolest of the year. The enhanced climatological zonal SST gradient in boreal summer reduces the convection over the eastern Pacific, which leads to a weakening of air–sea coupling strength. The minimum coupling strength during summer prevents SST anomalies from further development in the following season, which favors SST maximum during summer. In addition, enhanced zonal SST gradient and resultant thermocline shoaling over the eastern Pacific lead to excessive zonal advective feedback and thermocline feedback. Atmospheric damping is also weakened during boreal summer season. These changes due to feedback processes allow an excessive development of SST anomalies during the summer time, and lead to a summer peak of ENSO. The importance of basic state change for the ENSO phase-locking is also validated in a multi-model framework using the Coupled Model Intercomparison Project phase-3 archive. It is found that several of the climate models have the same problem in producing a summer peak of ENSO. Consistent with the simulations with different physical parameter values, these models have minimum air–sea coupling strength during the boreal summer season. Also, they have stronger climatological zonal SST gradient and shallower climatological thermocline depth over the eastern Pacific during the boreal summer season.

Complex Network Synchronizability: The Effect of Coupling Strength

In this paper, the effect of the coupling strength to the complex network synchronizability is investigated. For a given network with identical node dynamics, it is shown that the coupling strength among the nodes is one of key factors influencing the network synchronizability besides the network inner linking matrix and the eigenvalues of the network topological matrix. It is point that if the synchronized region is an unbounded sector, for achieving synchronizability, the coupling strength must be greater than or equal to the minimum coupling strength, and with the increasing of the coupling strength, network synchronizability is improved; if is a bounded sector, for achieving network synchronizability, the coupling strength must be in a certain range, and the larger coupling strength does not necessarily indicate better synchronizability.

How the self-coupled neuron can affect the chaotic synchronization of network

We have calculated 34 kinds of three-cell neuron networks’ minimum coupling strength, from the result; we find that a self-coupled neuron can have some effect on the synchronization of the network. The reason is the self-coupled neurons make the number of neurons looks “decrease”, and they decrease the coupling strength of the other neurons which are coupled with them.

Coherence transition of small Josephson junctions coupled to a single-mode resonant cavity: Connection to the Dicke model

We calculate the thermodynamic properties of a collection of N small Josephson junctions coupled to a single-mode resonant electromagnetic cavity, at finite temperature T, using several approaches. In the first approach, we include all the quantum-mechanical levels of the junction, but treat the junction-cavity interaction using a mean-field approximation developed previously for T = 0 . In the other approaches, the junctions are treated including only the two lowest energy levels per junction, but with two different Hamiltonians. The first of these maps onto the Dicke model of quantum optics. The second is a modified Dicke model which contains an additional XY-like coupling between the junctions. The modified Dicke model can be treated using a mean-field theory, which in the limit of zero XY coupling gives the solution of the Dicke model in the thermodynamic limit using Glauber coherent states to represent the cavity. In all cases, for an N-independent junction-cavity coupling, there is a critical junction number N above which there is a continuous transition from incoherence to coherence with decreasing T. If the coupling scales with N so as to give a well-behaved thermodynamic limit, there is a critical minimum coupling strength for the onset of coherence. In all three models, the cavity photon occupation numbers have a non-Bose distribution when the system is coherent.

Coupling effects on energy transduction in coupled polymer chains with perturbation of noise

Noise-assistant transduction was investigated in coupled polymer chains where one subsystem was exposed to environment noise. It was found that coupling could transfer oscillation from one subsystem disturbed by noise to the other not disturbed by noise and play a role of a noise filtering for the other. Then, a sort of coupling-induced synchronization was investigated as a function of noise intensity and coupling strength. In particular, we calculated the minimum coupling strength to reach synchronization and pointed out that noise dominated at small coupling strength, otherwise, coupling dominated.

Unilateral coupling synchronization of spatiotemporal chaos in the Bragg acousto-optic bistable system

The one- and two-dimensional coupled map lattices in nonlinear dynamic system are used as the system model. We report a method with which the spatiotemporal chaos can be synchronized using unilateral coupling in two Bragg acousto-optic bistable spatial extended systems. The synchronization can be realized by appropriately selecting the coupling strength and the equilibrium coefficient. By calculating the largest conditional Lyapunov exponent, we obtain the minimum coupling strength for achieving the synchronization and the functional relationship between the minimum coupling strength and the system parameters. Our simulation shows that the synchronization can also be realized under the influence of small random noise, so this method is robust.

Connection graph stability method for synchronized coupled chaotic systems

This paper elucidates the relation between network dynamics and graph theory. A new general method to determine global stability of total synchronization in networks with different topologies is proposed. This method combines the Lyapunov function approach with graph theoretical reasoning. In this context, the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph. In particular, the method is applied to the study of synchronization in rings of 2 K-nearest neighbor coupled oscillators. A rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators. This bound is explicitly linked with the average path length of the coupling graph. Contrary to the master stability function approach developed by Pecora and Carroll, the connection graph stability method leads to global stability of synchronization, and it permits not only constant, but also time-dependent interaction coefficients. In a companion paper (“Blinking model and synchronization in small-world networks with a time-varying coupling,” see this issue), this method is extended to the blinking model of small-world networks where, in addition to the fixed 2 K-nearest neighbor interactions, all the remaining links are rapidly switched on and off independently of each other.

Unilaterally coupled synchronization of spatiotemporal chaos

Based on a coupled map lattice model, we present an approach of unilateral coupling to implement the synchronization of spatiotemporal chaos. Our numerical simulations show that the two spatiotemporal systems can accurately become synchronized by appropriately selecting the coupling strength and the equilibrium coefficient. We calculate the largest conditional Lyapunov exponent，so as to give the minimum coupling strength that can achieve the synchronization，and the functional relationship between the minimum coupling strength and the system parameters. The simulation results also indicate that this method is robust against noise.

Interlayer exchange coupling versus ferromagnetic layer thickness in asymmetric Co/Ru/Co trilayer films

A dependence of the interlayer exchange coupling coefficient A12 on the Co layer thickness has been observed in asymmetric Co(32 Å)/Ru(tRu)/Co(t2) trilayer structures for several series within which tRu is a constant. As a function of the second Co layer thickness, the amplitude of A12 varies significantly but the sign of A12 does not change within each series. In both parallel and antiparallel coupled structures, the variation length Δt2 between the maximum and the minimum coupling strength is rather large (about 10 Å) and consistent from series to series.

Angular dependence of ferromagnetic resonance in exchange-coupled Co/Ru/Co trilayer structures.

The angular dependence of the dispersion relation is calculated for a system consisting of two ferromagnetic layers exchange coupled through a nonmagnetic spacer layer. Special attention has been focused on the resonance behavior in the unsaturated state of an antiparallel coupled system. The variation of both the acoustic mode and the optic mode with the external-field orientation is significantly modified from that in a noncoupled system and can be used to accurately evaluate the interlayer exchange-coupling strength A(${\mathrm{\ensuremath{\theta}}}_{12}$) as a function of the angle between the magnetization vectors in the two magnetic layers. Based on the exchange-coupled resonance theory, the angular dependence of ferromagnetic resonance (FMR) measurements has been performed on several series of symmetrical and asymmetrical Co/Ru/Co structures at X-band and K-band frequencies with the temperature ranging from 10 to 300 K. Only the bilinear exchange-coupling coefficient ${\mathit{A}}_{12}$ was observed in these systems. The biquadratic contribution is more than two orders of magnitude smaller than ${\mathit{A}}_{12}$. For the symmetrical Co(32 \AA{})/Ru(${\mathit{t}}_{\mathrm{Ru}}$)/Co(32 \AA{}) series, oscillatory interlayer exchange coupling was observed as a function of the Ru thickness ${\mathit{t}}_{\mathrm{Ru}}$. The oscillation period (\ensuremath{\sim}12 \AA{}) and phase do not vary with temperature. However, the oscillation amplitude is significantly enhanced at low temperatures, following roughly the relationship ${\mathit{A}}_{12}$\ensuremath{\propto}(T/${\mathit{T}}_{0}$)/sinh(T/${\mathit{T}}_{0}$) predicted by the theoretical models. For the asymmetrical Co(32 \AA{})/Ru(${\mathit{t}}_{\mathrm{Ru}}$)/Co(${\mathit{t}}_{2}$) structures, variation of the exchange coupling strength as a function of ${\mathit{t}}_{2}$ has also been observed for several series within which ${\mathit{t}}_{\mathrm{Ru}}$ is constant. The variation length \ensuremath{\Delta}${\mathit{t}}_{2}$ between maximum and minimum coupling strength is rather large (about 10 \AA{}) and consistent from series to series.

Chaos in extended linear arrays of Josephson weak links.

Extended linear arrays of interacting Josephson weak links are studied by numerical simulation using the resistively shunted junction model. The minimum coupling strength for chaotic behavior is determined as a function of the number of links. This strength is found to diminish steadily with increasing number, despite the inclusion of only nearest-neighbor interaction. The implications for Josephson technology are briefly discussed. Mathematically, the results are confirmation of the Ruelle-Takens scenario for chaos.