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Networks & Heterogeneous Media

2014 , Volume 9 , Issue 1

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Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Mattia Bongini and  Massimo Fornasier
2014, 9(1): 1-31 doi: 10.3934/nhm.2014.9.1 +[Abstract](89) +[PDF](2266.2KB)
Conditional self-organization and pattern-formation are relevant phenomena arising in biological, social, and economical contexts, and received a growing attention in recent years in mathematical modeling. An important issue related to optimal government strategies is how to design external parsimonious interventions, aiming at enforcing systems to converge to specific patterns. This is in contrast to other models where the players of the systems are allowed to interact freely and are supposed autonomously, either by game rules or by embedded decentralized feedback control rules, to converge to patterns. In this paper we tackle the problem of designing optimal centralized feedback controls for systems of moving particles, subject to mutual attraction and repulsion forces, and friction. Under certain conditions on the attraction and repulsion forces, if the total energy of the system, composed of the sum of its kinetic and potential parts, is below a certain critical threshold, then such systems are known to converge autonomously to the stable configuration of keeping confined and collision avoiding in space, uniformly in time. If the energy is above such a critical level, then the space coherence can be lost. We show that in the latter situation of lost self-organization, one can nevertheless steer the system to return to stable energy levels by feedback controls defined as the minimizers of a certain functional with $l_1$-norm penalty and constraints. Additionally we show that the optimal strategy in this class of controls is necessarily sparse, i.e., the control acts on at most one agent at each time. This is another remarkable example of how homophilious systems, i.e., systems where agents tend to be strongly more influenced by near agents than far ones, are naturally prone to sparse stabilization, explaining the effectiveness of parsimonious interventions of governments in societies.
Asymptotic synchronous behavior of Kuramoto type models with frustrations
Seung-Yeal Ha , Yongduck Kim and  Zhuchun Li
2014, 9(1): 33-64 doi: 10.3934/nhm.2014.9.33 +[Abstract](55) +[PDF](575.6KB)
We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization. We consider three Kuramoto-type models with frustrations. First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed. Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phase-locked states via the segregation stage. Finally, we consider a Kuramoto-type model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Lück and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained. Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.
Numerical network models and entropy principles for isothermal junction flow
Gunhild A. Reigstad
2014, 9(1): 65-95 doi: 10.3934/nhm.2014.9.65 +[Abstract](81) +[PDF](1803.6KB)
We numerically explore network models which are derived for the isothermal Euler equations. Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
    We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
    Numerical results are provided for two different network configurations and for three different network models. Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness. Analytical predictions for junctions connecting three pipe sections are verified for both network configurations. Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different cross-sectional area is shown.
Characteristic half space problem for the Broadwell model
Linglong Du
2014, 9(1): 97-110 doi: 10.3934/nhm.2014.9.97 +[Abstract](33) +[PDF](349.8KB)
We study an initial boundary value problem for the Broadwell model in half space. The Green's function for the initial boundary value problem is decomposed into two parts: one is the Green's function for the initial value problem, we call it the fundamental solution for the whole space; the other is the convolution of this fundamental solution with full boundary data. A new approach to obtain the full boundary data is established here. Finally, a nonlinear time-asymptotic stability of an equilibrium state is proved.
The derivation of continuum limits of neuronal networks with gap-junction couplings
Claudio Canuto and  Anna Cattani
2014, 9(1): 111-133 doi: 10.3934/nhm.2014.9.111 +[Abstract](61) +[PDF](2640.9KB)
We consider an idealized network, formed by $N$ neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for $N \to \infty$ of the resulting discrete model is thoroughly investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies for passing to the limit are analysed: i) a more conventional approach, based on a fixed nearest-neighbour connection topology accompanied by a suitable scaling of the diffusion coefficients; ii) a new approach, in which the number of connections to any given neuron varies with $N$ according to a precise law, which simultaneously guarantees the non-triviality of the limit and the locality of neuronal interactions. Both approaches yield in the limit a pde-based model, in which the distribution of action potential obeys a nonlinear reaction-convection-diffusion equation; convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are discussed, both theoretically and numerically.
Computational models for fluid exchange between microcirculation and tissue interstitium
Laura Cattaneo and  Paolo Zunino
2014, 9(1): 135-159 doi: 10.3934/nhm.2014.9.135 +[Abstract](237) +[PDF](1884.8KB)
The aim of this work is to develop a computational model able to capture the interplay between microcirculation and interstitial flow. Such phenomena are at the basis of the exchange of nutrients, wastes and pharmacological agents between the cardiovascular system and the organs. They are particularly interesting for the study of effective therapies to treat vascularized tumors with drugs. We develop a model applicable at the microscopic scale, where the capillaries and the interstitial volume can be described as independent structures capable to propagate flow. We facilitate the analysis of complex capillary bed configurations, by representing the capillaries as a one-dimensional network, ending up with a heterogeneous system characterized by channels embedded into a porous medium. We use the immersed boundary method to couple the one-dimensional with the three-dimensional flow through the network and the interstitial volume, respectively. The main idea consists in replacing the immersed network with an equivalent concentrated source term. After discussing the details for the implementation of a computational solver, we apply it to compare flow within healthy and tumor tissue samples.
Constant in two-dimensional $p$-compliance-network problem
Al-hassem Nayam
2014, 9(1): 161-168 doi: 10.3934/nhm.2014.9.161 +[Abstract](34) +[PDF](318.6KB)
We consider the problem of the minimization of the $p$-compliance functional where the control variables $\Sigma$ are taking among closed connected one-dimensional sets. We prove some estimate from below of the $p$-compliance functional in terms of the one-dimensional Hausdorff measure of $\Sigma$ and compute the value of a constant $\theta(p)$ appearing usually in $\Gamma$-limit functional of the rescaled $p$-compliance functional.
Motion of discrete interfaces in low-contrast periodic media
Giovanni Scilla
2014, 9(1): 169-189 doi: 10.3934/nhm.2014.9.169 +[Abstract](41) +[PDF](434.4KB)
We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional low-contrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuum analysis. As in a recent paper by Braides and Scilla dealing with high-contrast periodic media, we give an example showing that in general the effective motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. We show that there exists a critical value $\widetilde{\delta}$ of the contrast parameter $\delta$ above which the discrete motion is constrained and coincides with the high-contrast case. If $\delta<\widetilde{\delta}$ we have a new pinning threshold and a new effective velocity both depending on $\delta$. We also consider the case of non-uniform inclusions distributed into periodic uniform layers.
A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function
Boris Muha
2014, 9(1): 191-196 doi: 10.3934/nhm.2014.9.191 +[Abstract](97) +[PDF](317.1KB)
The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.

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