October  2018, 23(8): 3415-3426. doi: 10.3934/dcdsb.2018283

Chaotic dynamics in a transport equation on a network

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

Received  June 2017 Revised  April 2018 Published  August 2018

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.

Citation: Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
References:
[1]

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show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.   Google Scholar

[4]

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.  Google Scholar

[5]

J. BanasiakA. 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.  Google Scholar

[6]

F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge University Press, 2009. Google Scholar

[7]

X. BarrachinaJ. A. ConejeroM. 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.  Google Scholar

[8]

J. A. ConejeroM. 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.  Google Scholar

[9]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356.  doi: 10.1007/s00233-007-9036-2.  Google Scholar

[10]

B. Dorn, Flows in Infinite Networks- a Semigroup Approach, Verlag Dr. Hut, München, 2009. Google Scholar

[11]

K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations: Graduate Texts in Mathematics, Springer-Verlag, New York, vol 194, 2000. Google Scholar

[12]

K-G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer-Verlag, London, 2011. Google Scholar

[13]

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.  Google Scholar

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