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The Logistic Map and the Route to Chaos pp 77–97Cite as

Coherence in Complex Networks of Oscillators

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Part of the book series: Understanding Complex Systems ((UCS))

Summary

We study fully synchronized (coherent) states in complex networks of chaotic oscillators, reviewing the analytical approach of determining the stability conditions for synchronizability and comparing them with numerical criteria. As an example, we present detailed results for networks of chaotic logistic maps having three different scale-free topologies: random scale-free topology, deterministic pseudo-fractal scale-free network and Apollonian network. For random scale-free topology we find that the lower boundary of the synchronizability region scales approximately as k −μ, where k is the outgoing connectivity and μ depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling is heterogeneous, namely when it is proportional to some power of the neighbour connectivity. In all cases, stability conditions are determined from the eigenvalue spectrum of the Laplacian matrix and agree well with numerical results based on histograms of coherent states in parameter space. Additionally, we show that almost everywhere in the synchronizability region the basin of attraction of the coherent states fills the entire phase space, and that the transition to coherence is of first-order.

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Author information

Authors and Affiliations

  1. ICP, Universität Stuttgart, Pfaffenwaldring 27, D-70569, Stuttgart, Germany

    Pedro G. Lind, Jason A.C. Gallas & Hans J. Herrmann

  2. Centro de Física Teórica e Computacional, Av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal

    Pedro G. Lind

  3. Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970, Porto Alegre, Brazil

    Jason A.C. Gallas

  4. Departamento de Física, Universidade Federal do Ceará, 60451-970, Fortaleza, Brazil

    Hans J. Herrmann

Authors
  1. Pedro G. Lind
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  2. Jason A.C. Gallas
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  3. Hans J. Herrmann
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Editor information

Editors and Affiliations

  1. Institut de Physique, Université de Liège, B5, Sart Tilman, 4000, Liège, Belgium

    Marcel Ausloos

  2. Royal Military Academy/ELEC, Avenue de la, Renaissancelaan 30, 1000, Brussels, Belgium

    Michel Dirickx

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© 2006 Springer-Verlag Berlin Heidelberg

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Lind, P.G., Gallas, J.A., Herrmann, H.J. (2006). Coherence in Complex Networks of Oscillators. In: Ausloos, M., Dirickx, M. (eds) The Logistic Map and the Route to Chaos. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-32023-7_5

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