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Observation and inverse problems in coupled cell networks

25 pages
Observation and inverse problems in coupled cell networks Romain Joly March 2011 Abstract A coupled cell network is a model for many situations such as food webs in ecosys- tems, cellular metabolism, economical networks... It consists in a directed graph G, each node (or cell) representing an agent of the network and each directed arrow rep- resenting which agent acts on which one. It yields a system of differential equations x˙(t) = f(x(t)), where the component i of f depends only on the cells xj(t) for which the arrow j ? i exists in G. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (os- cillations, stabilisation etc.) by observing only one of the cells? We show that the natural observation properties holds for almost all the interactions f . Key words: coupled cell networks, observability, inverse problems, genericity, tran- versality theorems. AMS subject classification: 93B07, 34C25, 34H15, 92B25. 1 Introduction The coupled cell networks. In the recent years, the mathematical study of coupled cell networks has been quickly developing. It combines several interests: it is strongly related with applications and real phenomena, the setting is very simple and it leads to a rich class of mathematically inter- esting problems.

  • can also

  • also stop

  • x1 ?

  • interaction

  • admissible vector fields

  • many concrete

  • networks

  • coupled cell

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