Chaotic dynamics in a transport equation on a network

Proscovia Namayanja

doi:10.3934/dcdsb.2018283

Article Contents

2018, Volume 23, Issue 8: 3415-3426. Doi: 10.3934/dcdsb.2018283

This issue Previous Article Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection Next Article Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces

Chaotic dynamics in a transport equation on a network

Proscovia Namayanja

School of Mathematics, Statistics & Computer Sciences, University of KwaZulu-Natal, Private Bax X54001, Durban 4001, South Africa

Received: May 31, 2017

Revised: March 31, 2018

Early access: August 2018

Published: August 2018

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

We show that for a system of transport equations defined on an infinite network, the semigroup generated is hypercyclic if and only if the adjacency matrix of the line graph is also hypercyclic. We further show that there is a range of parameters for which a transport equation on an infinite network with no loops is chaotic on a subspace $X_e$ of the weighted Banach space $\ell^1_s$. We relate these results to Banach-space birth-and-death models in literature by showing that when there is no proliferation, the birth-and-death model is also chaotic in the same subspace $X_e$ of $\ell^1_s$. We do this by noting that the eigenvalue problem for the birth-and-death model is in fact an eigenvalue problem for the adjacency matrix of the line graph (of the network on which the transport problem is defined) which controls the dynamics of the the transport problem.

Keywords:

Mathematics Subject Classification: Primary: 47A16, 47A10, 35A09; Secondary: 05C20, 05C21, 15B48.

Citation:

Full Text(HTML)

Figure 1. A graph of the Birth-death model with no proliferation

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	J. Banasiak , Chaos in Kolmogorov systems with proliferation- general criteria and applications, J.Math. Anal. Appl., 378 (2011) , 89-97. doi: 10.1016/j.jmaa.2010.12.054.
	J. Banasiak and M. Lachowicz , Topological chaos for birth-and-death-type models with proliferation, Math. Models Methods Appl. Sci., 12 (2002) , 755-775. doi: 10.1142/S021820250200188X.
	J. Banasiak and M. Moszynski , Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete and continuous dynamical systems, 29 (2011) , 67-79.
	J. Banasiak and P. Namayanja , Asymptotic behaviour of flows on reducible networks, Networks and Heterogeneous Media, 9 (2014) , 197-216. doi: 10.3934/nhm.2014.9.197.
	J. Banasiak , A. Falkiewicz and P. Namayanja , Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. models methods Appl. sci., 26 (2016) , 215-247. doi: 10.1142/S0218202516400017.
	F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge University Press, 2009.
	X. Barrachina , J. A. Conejero , M. Murillo-Arcila and J. B. Seoane-Sepulveda , Distributional chaos for the forward and backward control traffic model, Linear Algebra and its Applications, 479 (2015) , 202-215. doi: 10.1016/j.laa.2015.04.010.
	J. A. Conejero , M. Murillo-Arcila and J. B. Seoane-Sepulveda , Linear chaos for the quick thinking car-driver model, Semigroup Forum, 92 (2016) , 486-493. doi: 10.1007/s00233-015-9704-6.
	B. Dorn , Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008) , 341-356. doi: 10.1007/s00233-007-9036-2.
	B. Dorn, Flows in Infinite Networks- a Semigroup Approach, Verlag Dr. Hut, München, 2009.
	K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations: Graduate Texts in Mathematics, Springer-Verlag, New York, vol 194, 2000.
	K-G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer-Verlag, London, 2011.
	M. Kramar and E. Sikolya , Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005) , 139-162. doi: 10.1007/s00209-004-0695-3.