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Networks & Heterogeneous Media

March 2017 , Volume 12 , Issue 1

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Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators
Seung-Yeal Ha, Jaeseung Lee and Zhuchun Li
2017, 12(1): 1-24 doi: 10.3934/nhm.2017001 +[Abstract](624) +[HTML](2) +[PDF](866.2KB)
Abstract:

We study the emergence of local exponential synchronization in an ensemble of generalized heterogeneous Kuramoto oscillators with different intrinsic dynamics. In the classic Kuramoto model, intrinsic dynamics are given by the Kronecker flow with constant natural frequencies. We generalize the constant natural frequencies to smooth functions that depend on the state and time so that it can describe a more realistic situation arising from neuroscience. In this setting, the ensemble of generalized Kuramoto oscillators loses its synchronization even when the coupling strength is large. This leads to the study of a concept of "relaxed" synchronization, which is called "practical synchronization" in literature. In this new concept of "weak" synchronization, the phase diameter of the entire ensemble is uniformly bounded by some constant inversely proportional to the coupling strength. We focus on the complete synchronizability of a subensemble consisting of generalized Kuramoto oscillators with the same intrinsic dynamics; moreover, we provide several sufficient frameworks leading to local exponential synchronization of each homogeneous subensemble, although the whole ensemble is not fully synchronized. This is a generalization of an earlier analytical result regarding practical synchronization. We also provide several numerical simulations and compare them with analytical results.

Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling
Hirotada Honda
2017, 12(1): 25-57 doi: 10.3934/nhm.2017002 +[Abstract](459) +[HTML](3) +[PDF](447.5KB)
Abstract:

In this paper, we focus on the global-in-time solvability of the Kuramoto-Sakaguchi equation under non-local coupling. We further study the nonlinear stability of the trivial stationary solution in the presence of sufficiently large diffusivity, and the existence of the solution under vanishing diffusion.

Modelling heterogeneity and an open-mindedness social norm in opinion dynamics
Clinton Innes, Razvan C. Fetecau and Ralf W. Wittenberg
2017, 12(1): 59-92 doi: 10.3934/nhm.2017003 +[Abstract](456) +[HTML](4) +[PDF](1434.5KB)
Abstract:

We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.

A discrete Hughes model for pedestrian flow on graphs
Fabio Camilli, Adriano Festa and Silvia Tozza
2017, 12(1): 93-112 doi: 10.3934/nhm.2017004 +[Abstract](484) +[HTML](1) +[PDF](1093.6KB)
Abstract:

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.

A mathematical framework for delay analysis in single source networks
Samitha Samaranayake, Axel Parmentier, Ethan Xuan and Alexandre Bayen
2017, 12(1): 113-145 doi: 10.3934/nhm.2017005 +[Abstract](319) +[HTML](1) +[PDF](1271.8KB)
Abstract:

This article presents a mathematical framework for modeling heterogeneous flow networks with a single source and multiple sinks with no merging. The traffic is differentiated by the destination (i.e. Lagrangian flow) and different flow groups are assumed to satisfy the first-in-first-out (FIFO) condition at each junction. The queuing in the network is assumed to be contained at each junction node and spill-back to the previous junction is ignored. We show that this model leads to a well-posed problem for computing the dynamics of the system and prove that the solution is unique through a mathematical derivation of the model properties. The framework is then used to analytically prescribe the delays at each junction of the network and across any sub-path, which is the main contribution of the article. This is a critical requirement when solving control and optimization problems over the network, such as system optimal network routing and solving for equilibrium behavior. In fact, the framework provides analytical expressions for the delay at any node or sub-path as a function of the inflow at any upstream node. Furthermore, the model can be solved numerically using a very simple and efficient feed forward algorithm. We demonstrate the versatility of the framework by applying it to two example networks, a single path of multiple bottlenecks and a diverge junction with complex junction dynamics.

An improved homogenization result for immiscible compressible two-phase flow in porous media
Brahim Amaziane, Leonid Pankratov and Andrey Piatnitski
2017, 12(1): 147-171 doi: 10.3934/nhm.2017006 +[Abstract](486) +[HTML](4) +[PDF](544.7KB)
Abstract:

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

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