Generalized network transport and Euler-Hille formula

Jacek Banasiak; Aleksandra Puchalska

doi:10.3934/dcdsb.2018185

Article Contents

2018, Volume 23, Issue 5: 1873-1893. Doi: 10.3934/dcdsb.2018185

This issue Previous Article Preface: Special issue on dynamical systems on graphs Next Article On arbitrarily long periodic orbits of evolutionary games on graphs

Generalized network transport and Euler-Hille formula

Jacek Banasiak^1,2, , and
Aleksandra Puchalska^2,3,

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
2.
Institute of Mathematics, Łódź University of Technology, Łódź, Poland
3.
Advanced System Analysis Program, International Institute for Applied System Analysis, Laxenburg, Austria

* Corresponding author: Jacek Banasiak
* Corresponding author: Jacek Banasiak

Received: April 2017

Revised: August 2017

Early access: May 2018

Published: May 2018

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Abstract

In this article we consider asymptotic properties of network flow models with fast transport along the edges and explore their connection with an operator version of the Euler formula for the exponential function. This connection, combined with the theory of the regular convergence of semigroups, allows for proving that for fast transport along the edges and slow rate of redistribution of the flow at the nodes, the network flow semigroup (or its suitable projection) can be approximated by a finite dimensional dynamical system related to the boundary conditions at the nodes of the network. The novelty of our results lies in considering more general boundary operators than that allowed for in previous papers.

Keywords:

Mathematics Subject Classification: Primary: 47D06, 35B25; Secondary: 35L03, 347D62, 05C21, 15A16.

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Figure 1. Commutativity of the aggregation diagram

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Figure 2. The graph G representing the canal network in Example 1

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Figure 3. The line graph of the graph shown on Fig. 2

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Figure 4. Graphical representation of the Kimmel–Stivers model

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Figure 5. Kimmel–Stievers model with vital dynamics

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Figure 6. Discrete Lebowitz–Rubinow–Rotenberg model

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