Mean field theory of assortative networks of phase oscillators

Employing the Kuramoto model as an illustrative example, we show how the use of the mean field approximation can be applied to large networks of phase oscillators with assortativity. We then use the ansatz of Ott and Antonsen [Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a system of ordinary differential equations. The resulting formulation is illustrated by application to a network Kuramoto problem with degree assortativity and correlation between the node degrees and the natural oscillation frequencies. Good agreement is found between the solutions of the reduced set of ordinary differential equations obtained from our theory and full simulations of the system. These results highlight the ability of our method to capture all the phase transitions (bifurcations) and system attractors. One interesting result is that degree assortativity can induce transitions from a steady macroscopic state to a temporally oscillating macroscopic state through both (presumed) Hopf and SNIPER (saddle-node, infinite period) bifurcations. Possible use of these techniques to a broad class of phase oscillator network problems is discussed.Comment: 8 pages, 7 figure

Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans

The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes -- or phase reversals -- low-periodic dynamics prevail, while away from the nodes the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model

Synchronization in large directed networks of coupled phase oscillators

We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider extensions to networks with mixed positive/negative coupling strengths. We compare our theory with numerical simulations and find good agreement

Downlink Coverage Analysis in a Heterogeneous Cellular Network

In this paper, we consider the downlink signal-to-interference-plus-noise ratio (SINR) analysis in a heterogeneous cellular network with K tiers. Each tier is characterized by a base-station (BS) arrangement according to a homogeneous Poisson point process with certain BS density, transmission power, random shadow fading factors with arbitrary distribution, arbitrary path-loss exponent and a certain bias towards admitting the mobile-station (MS). The MS associates with the BS that has the maximum SINR under the open access cell association scheme. For such a general setting, we provide an analytical characterization of the coverage probability at the MS.Comment: 6 pages, 5 figures, submitted to IEEE Globecom 2012 - Wireless Communications Symposium on Apr 2, 201

Onset of Synchronization in the Disordered Hamiltonian Mean-Field Model

We study the Hamiltonian mean field (HMF) model of coupled Hamiltonian rotors with a heterogeneous distribution of moments of inertia and coupling strengths. We show that when the parameters of the rotors are heterogeneous, finite-size fluctuations can greatly modify the coupling strength at which the incoherent state loses stability by inducing correlations between the momenta and parameters of the rotors. When the distribution of initial frequencies of the oscillators is sufficiently narrow, an analytical expression for the modification in critical coupling strength is obtained that confirms numerical simulations. We find that heterogeneity in the moments of inertia tends to stabilize the incoherent state, while heterogeneity in the coupling strengths tends to destabilize the incoherent state. Numerical simulations show that these effects disappear for a wide, bimodal frequency distribution