Synchronizability of chaotic logistic maps in delayed complex networks

We study a network of coupled logistic maps whose interactions occur with a certain distribution of delay times. The local dynamics is chaotic in the absence of coupling and thus the network is a paradigm of a complex system. There are two regimes of synchronization, depending on the distribution of delays: when the delays are sufficiently heterogeneous the network synchronizes on a steady-state (that is unstable for the uncoupled maps); when the delays are homogeneous, it synchronizes in a time-dependent state (that is either periodic or chaotic). Using two global indicators we quantify the synchronizability on the two regimes, focusing on the roles of the network connectivity and the topology. The connectivity is measured in terms of the average number of links per node, and we consider various topologies (scale-free, small-world, star, and nearest-neighbor with and without a central hub). With weak connectivity and weak coupling strength, the network displays an irregular oscillatory dynamics that is largely independent of the topology and of the delay distribution. With heterogeneous delays, we find a threshold connectivity level below which the network does not synchronize, regardless of the network size. This minimum average number of neighbors seems to be independent of the delay distribution. We also analyze the effect of self-feedback loops and find that they have an impact on the synchronizability of small networks with large coupling strengths. The influence of feedback, enhancing or degrading synchronization, depends on the topology and on the distribution of delays.     

Spectral universality of phase synchronization in non-identical oscillator networks

We employ a spectral decomposition method to analyze synchronization of a non-identical oscillator network. We study the case that a small parameter mismatch of oscillators is characterized by one parameter and phase synchronization is observed. We derive a linearized equation for each eigenmode of the coupling matrix. The parameter mismatch is reflected on inhomogeneous term in the linearized equation. We find that the oscillation of each mode is essentially characterized only by the eigenvalue of the coupling matrix with a suitable normalization. We refer to this property as spectral universality, because it is observed irrespective of network topology. Numerical results in various network topologies show good agreement with those based on linearized equation. This universality is also observed in a system driven by additive independent Gaussian noise.     

Metastability and functional integration in anisotropically coupled map lattices

Metastability is a property of systems composed of many interacting parts wherein the parts exhibit simultaneously a tendency to function autonomously (local segregation) and a tendency to cooperate (global integration). We study anisotropically coupled map lattices and discover that for specific values of the coupling control parameters the entire system transits to a metastable regime. We show that this regime manifests a quasi-stable state in which the system can flexibly switch to another such state. We briefly discuss the relevance of our findings for information processing, functional integration, metastability in the brain, and phase transitions in complex systems.     

The study of banded spherulite patterns by coupled logistic map lattice model

Banded spherulite patterns are simulated in two dimensions by means of a coupled logistic map lattice model. Both target pattern and spiral pattern which have been proved to be existent experimentally in banded spherulite are obtained by choosing suitable parameters in the model. The simulation results also indicate that the band spacing is decreased with the increase of parameter μ in the logistic map and increased with the increase of the coupling parameter ε, which is quite similar to the results in some experiments. Moreover, the relationship between the parameters and the corresponding patterns is obtained, and the target patterns and spiral patterns are distinguished for a given group of initial values, which may guide the study of banded spherulite.     

Synchronization,stickiness effects and intermittent oscillations in coupled nonlinear stochastic networks

Long distance reactive and diffusive coupling is introduced in a spatially extended nonlinear stochastic network of interacting particles. The network serves as a substrate for Lotka-Volterra dynamics with 3rd order nonlinearities. If the network includes only local nearest neighbour interactions, the system organizes into a number of local asynchronous oscillators. It is shown that (a) Introduction of all-to-all coupling in the network leads the system into global, center-type, conservative oscillations as dictated by the mean-field dynamics. (b) Introduction of reactive coupling to the network leads the system to intermittent oscillations where the trajectories stick for short times in bounded regions of space, with subsequent jumps between different bounded regions. (c) Introduction of diffusive coupling to the system does not alter the dynamics for small values of the diffusive coupling p_diff, while after a critical value p_diff ^c the system synchronizes into a limit cycle with specific frequency, deviating non-trivially from the mean-field center-type behaviour. The frequency of the limit cycle oscillations depends on the reaction rates and on the diffusion coupling. The amplitude σ of the limit cycle depends on the control parameter p_diff.     

Analytical treatment of synchronization of spin-torque oscillators by microwave magnetic fields

An analytical approach is presented for the study of synchronization effects in spin-transfer-driven nanomagnets subjected to radio-frequency magnetic fields. The conditions are derived and discussed under which the current-induced magnetization precession is synchronized by the radio-frequency field. Exact analytical results are obtained for the case when the problem exhibits uniaxial symmetry around the axis perpendicular to the device plane. It is demonstrated that the magnetization dynamics under nonzero current and nonzero rf field is identical in structure to that under zero current. On this basis, analytical predictions are obtained for: the existence of phase-locking between current-induced magnetization precession and rf field oscillations; the frequency pulling effect in proximity of phase locking; the occurrence of hysteresis effects in phase-locking as a function of the spin-polarized current. The proposed approach is valid for arbitrary rf field amplitude and current intensity.     

Chaos in coupled lasers with low-frequency modulation

The dynamics of the array consisting of two coupled solid-state lasers with frequency modulations was researched numerically. Array intensity’s chaotic behavior is predicted when the modulation frequency is low. The physical mechanism of the chaos is discussed here qualitatively. It was found that not only a harmonic resonance itself but also the duration of the harmonic resonance long enough is needed for the appearance of the chaos.     

Consensus on de Bruijn graphs

We study the consensus dynamics with or without time-delays on directed and undirected de Bruijn graphs. Our results show that consensus on an undirected de Bruijn graph has a lower converging speed and larger time-delay tolerance in comparison with that on an undirected scale-free network. Although there is not much difference between the eigenvalue ratios of the two undirected networks, we found that their dynamical properties are remarkably different; consequently, it is seemingly more informative to consider the second smallest and the largest eigenvalues separately rather than considering their ratio in the study of synchronization of a coupled oscillators network. Moreover, our study on directed de Bruijn graphs reveals that properly setting directions on edges can improve the converging speed and time-delay tolerance simultaneously.     

Multivariate analysis of spatially heterogeneous phase synchronisation in complex systems: application to self-organised control of material flows in networks

Networks of interacting components are a class of complex systems that has attracted considerable interest over the last decades. In particular, if the dynamics of the autonomous components is characterised by an oscillatory behaviour, different types of synchronisation can be observed in dependence on the type and strength of interactions. In this contribution, we study the transition from non-synchronised to synchronised phase dynamics in complex networks. The most common approach to quantify the degree of phase synchronisation in such systems is the consideration of measures of phase coherence which are averaged over all pairs of interacting components. However, this approach implicitly assumes a spatially homogeneous synchronisation process, which is typically not present in complex networks. As a potential alternative, two novel methods of multivariate phase synchronisation analysis are considered: synchronisation cluster analysis (SCA) and the linear variance decay (LVD) dimension method. The strengths and weaknesses of the traditional as well as both new approaches are briefly illustrated for a Kuramoto model with long-range coupling. As a practical application, we study how spatial heterogeneity influences the transition to phase synchronisation in traffic networks where intersecting material flows are subjected to a self-organised decentralised control. We find that the network performance and the degree of phase synchronisation are closely related to each other and decrease significantly in the case of structural heterogeneities. The influences of the different parameters of our control approach on the synchronisation process are systematically studied, yielding a sequence of Arnold tongues which correspond to different locking modes.     

Communities recognition in the Chesapeake Bay ecosystem by dynamical clustering algorithms based on different oscillators systems

We have recently introduced [Phys. Rev. E 75, 045102(R) (2007); AIP Conference Proceedings 965, 2007, p. 323] an efficient method for the detection and identification of modules in complex networks, based on the de-synchronization properties (dynamical clustering) of phase oscillators. In this paper we apply the dynamical clustering tecnique to the identification of communities of marine organisms living in the Chesapeake Bay food web. We show that our algorithm is able to perform a very reliable classification of the real communities existing in this ecosystem by using different kinds of dynamical oscillators. We compare also our results with those of other methods for the detection of community structures in complex networks.     

Stochastic resonance in finite arrays of bistable elements with local coupling

In this article, we investigate the stochastic resonance (SR) effect in a finite array of noisy bistable systems with nearest-neighbor coupling driven by a weak time-periodic driving force. The array is characterized by a collective variable. By means of numerical simulations, the signal-to-noise ratio (SNR) and the gain are estimated as functions of the noise and the interaction coupling strength. A strong enhancement of the SR phenomenon for this collective variable in comparison with SR in single unit bistable systems is observed. Gains larger than unity are obtained for some parameter values and multi-frequency driving forces, indicating that the system is operating in a non-linear regime albeit the smallness of the driving amplitude. The large SNR values observed are basically due to the fact that the output fluctuations are small and short lived, in comparison with their typical values in a linear regime. A non-monotonic behavior of the SNR with the coupling strength is also obtained.     

The effect of low-frequency oscillations on cardio-respiratory synchronization

We show that the transitions which occur between close orders of synchronization in the cardiorespiratory system are mainly due to modulation of the cardiac and respiratory processes by low-frequency components. The experimental evidence is derived from recordings on healthy subjects at rest and during exercise. Exercise acts as a perturbation of the system that alters the mean cardiac and respiratory frequencies and changes the amount of their modulation by low-frequency oscillations. The conclusion is supported by numerical evidence based on a model of phase-coupled oscillators, with white noise and lowfrequency noise. Both the experimental and numerical approaches confirm that low-frequency oscillations play a significant role in the transitional behavior between close orders of synchronization.     

Global firing induced by network disorder in ensembles of active rotators

We study the influence of repulsive interactions on an ensemble of coupled excitable rotators. We find that a moderate fraction of repulsive interactions can trigger global firing of the ensemble. The regime of global firing, however, is suppressed in sufficiently large systems if the network of repulsive interactions is fully random, due to self-averaging in its degree distribution. We thus introduce a model of partially random networks with a broad degree distribution, where self-averaging due to size growth is absent. In this case, the regime of global firing persists for large sizes. Our results extend previous work on the constructive effects of diversity in the collective dynamics of complex systems.     

Global synchronization in arrays of coupled networks with one single time-varying delay coupling

Global synchronization in arrays of coupled networks with one single time-varying delay coupling is investigated in this Letter. A general linear coupled network with a time-varying coupling delay is proposed and its global synchronization is further discussed. Some sufficient criteria are derived based on Lyapunov functional and linear matrix inequality (LMI). It is shown that under one single delay coupling, the synchronized state changes, which is different from the conventional synchronized solution. Moreover, the degree of the nodes and the inner delayed coupling matrix play key roles in the synchronized state. In particular, the derivative of the time-varying delay can be any given value. Finally, numerical simulations are given to illustrate the theoretical results.     

Stochastic resonance in bistable confining potentials

We study the effects of the confining conditions on the occurrence of stochastic resonance (SR) in continuous bistable systems. We model such systems by means of double-well potentials that diverge like |x|^q for |x|↦∞. For super-harmonic (hard) potentials with q > 2 the SR peak sharpens with increasing q, whereas for sub-harmonic (soft) potentials, q < 2, it gets suppressed.     

Synchronization,zero-resistance states and rotating Wigner crystal

We show, in a framework of a classical nonequilibrium model, that rotational angles of electrons moving in two dimensions (2D) in a perpendicular magnetic field can be synchronized by an external microwave field whose frequency is close to the Larmor frequency. The synchronization eliminates collisions between electrons and thus creates a regime with zero diffusion corresponding to the zero-resistance states observed in experiments with high mobility 2D electron gas (2DEG). For long range Coulomb interactions electrons form a rotating hexagonal Wigner crystal. Possible relevance of this effect of synchronization-induced self-assembly for planetary rings is discussed.     

Effect of delay on phase locking in a pulse coupled neural network

Using a slightly simplified version of the integrate and fire model of a neural network with delay, I study the stability of the phase-locked state dependent on the coupling between the neurons and especially on a delay time. The coupling between neurons may be arbitrary. It is shown that the phase-locked state becomes less stable with increasing delay and that relaxation oscillations occur. Received 28 December 1999 and Received in final form 13 June 2000     

Interplay of noise and coupling in heterogeneous ensembles of phase oscillators

We study the effects of noise on the collective dynamics of an ensemble of coupled phase oscillators whose natural frequencies are all identical, but whose coupling strengths are not the same all over the ensemble. The intensity of noise can also be heterogeneous, representing diversity in the individual responses to external fluctuations. We show that the desynchronization transition induced by noise may be completely suppressed, even for arbitrarily large noise intensities, is the distribution of coupling strengths decays slowly enough for large couplings. Equivalently, if the response to noise of a sufficiently large fraction of the ensemble is weak enough, desynchronization cannot occur. The two effects combine with each other when the response to noise and the coupling strength of each oscillator are correlated. This combination is quantitatively characterized and illustrated with explicit examples.