Abstract
Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the last decades there has been an intensive research in the geometrical characterization of global attractors. However, there still exists a weak connection between the asymptotic dynamics of a complex network and the structure of associated global attractors. In this paper we show that, in order to analyze the long-time behavior of the dynamics on a complex network, it is the topological and geometrical structure of the attractor the subject to take into account. In fact, given a complex network, a global attractor can be understood as the new attracting complex network which is really describing and determining the forwards dynamics of the phenomena. We illustrate our discussion with models of differential equations related to mutualistic complex networks in Economy and Ecology.
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Acknowledgements
J.A. Langa is partially supported by FEDER and Ministerio de Economía y Competitividad grant # MTM2015-63723-P, Junta de Andalucía under Proyecto de Excelencia FQM-1492 and Brazilian-European partnership in Dynamical Systems (BREUDS) from the FP7-IRSES grant of the European Union.
A. Suárez is partially supported by FEDER and Ministerio de Economía y Competitividad grant # MTM2012-31304. This Chapter is based upon work from COST Action ISCH COST Action IS1104 “The EU in the new complex geography of economic systems: models, tools and policy evaluation”, supported by COST (European Cooperation in Science and Technology) www.cost.eu.
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Guerrero, G., Langa, J.A., Suárez, A. (2016). Attracting Complex Networks. In: Commendatore, P., Matilla-García, M., Varela, L., Cánovas, J. (eds) Complex Networks and Dynamics. Lecture Notes in Economics and Mathematical Systems, vol 683. Springer, Cham. https://doi.org/10.1007/978-3-319-40803-3_12
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