Abstract
This is a fundamental chapter of the book. It deals with networks, which are here considered as graphs, and is built on the theory developed in the previous chapter, on matrices.
Although, the dynamical networks described in Chap. 3 are richer objects than their weighted adjacency matrices, the latter still carry the most important information about a dynamical network. Indeed, from a theoretical point of view, a network’s weighted adjacency matrix describes a linearization of the network’s dynamics, which in applications is often the only network information available. In fact, it is not uncommon to have only the unweighted adjacency matrix of a network.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
V. Afraimovich, L. Bunimovich, Dynamical networks: interplay of topology, interactions, and local dynamics. Nonlinearity 20, 1761–1771 (2007)
L.A. Bunimovich, B.Z. Webb, Isospectral graph transformations, spectral equivalence, and global stability of dynamical networks. Nonlinearity 25, 211–254 (2012)
L.A. Bunimovich, B.Z. Webb, Restrictions and stability of time-delayed dynamical networks. Nonlinearity 26, 2131–2156 (2013)
S. Chena, W. Zhaoa, Y. Xub, New criteria for globally exponential stability of delayed Cohen–Grossberg neural network. Math. Comput. Simul. 79, 1527–1543 (2009)
M. Cohen, S. Grossberg, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern. SMC-13, 815–821 (1983)
R. Horn, C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1990)
L. Tao, W. Ting, F. Shumin, Stability analysis on discrete-time Cohen–Grossberg neural networks with bounded distributed delay, in Proceedings of the 30th Chinese Control Conference, Yantai, China, 22–24 July 2011
L. Wang, Stability of Cohen–Grossberg neural networks with distributed delays. Appl. Math. Comput. 160, 93–110 (2005)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bunimovich, L., Webb, B. (2014). Stability of Dynamical Networks. In: Isospectral Transformations. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1375-6_3
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1375-6_3
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1374-9
Online ISBN: 978-1-4939-1375-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)