Coherent regimes of globally coupled dynamical systems

The paper presents a method by which the mean field dynamics of a population of dynamical systems with parameter diversity and global coupling can be described in terms of a few macroscopic degrees of freedom. The method applies to populations of any size and functional form in the region of coherence. It requires linear variation or a narrow distribution for the dispersed parameter. Although being an approximation, the method allows us to quantitatively study the collective regimes that arise as a result of diversity and coupling and to interpret the transitions among these regimes as bifurcations of the effective macroscopic degrees of freedom. To illustrate, the phenomenon of oscillator death and the route to full locking are examined for chaotic oscillators with time scale mismatch.Comment: 5 pages, 3 figure

Low-dimensional chaos in populations of strongly-coupled noisy maps

We characterize the macroscopic attractor of infinite populations of noisy maps subjected to global and strong coupling by using an expansion in order parameters. We show that for any noise amplitude there exists a large region of strong coupling where the macroscopic dynamics exhibits low-dimensional chaos embedded in a hierarchically-organized, folded, infinite-dimensional set. Both this structure and the dynamics occuring on it are well-captured by our expansion. In particular, even low-degree approximations allow to calculate efficiently the first macroscopic Lyapunov exponents of the full system.Comment: 16 pages, 9 figures. Progress of Theoretical Physics, to appea

The interaction of thin-film flow, bacterial swarming and cell differentiation in colonies of Serratia liquefaciens

The rate of expansion of bacterial colonies of S. liquefaciens is investigated in terms of a mathematical model that combines biological as well as hydrodynamic processes. The relative importance of cell differentiation and production of an extracellular wetting agent to bacterial swarming is explored using a continuum representation. The model incorporates aspects of thin film flow with variable suspension viscosity, wetting, and cell differentiation. Experimental evidence suggests that the bacterial colony is highly sensitive to its environment and that a variety of mechanisms are exploited in order to proliferate on a variety of surfaces. It is found that a combination of effects are required to reproduce the variation of bacterial colony motility over a large range of nutrient availability and medium hardness

Cluster and group synchronization in delay-coupled networks

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.Comment: 11 pages, 7 figure

The onset of synchronization in large networks of coupled oscillators

We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition, and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix. We show how, with an additional assumption, a mean field approximation recently proposed is recovered from our results. We test our theory with numerical simulations, and find that it describes the transition when our assumptions are satisfied. We find that our theory describes the transition well in situations in which the mean field approximation fails. We study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase.Comment: To appear in PRE; Added an Appendix, a reference, modified two figures and improved the discussion of the range of validity of perturbative approache

Enhance synchronizability via age-based coupling

In this brief report, we study the synchronization of growing scale-free networks. An asymmetrical age-based coupling method is proposed with only one free parameter $\alpha$. Although the coupling matrix is asymmetric, our coupling method could guarantee that all the eigenvalues are non-negative reals. The eigneratio R will approach to 1 in the large limit of $\alpha$.Comment: 3 pages, 1 figur