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Chaotic dynamics in a transport equation on a network

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  • 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.

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


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

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