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Complex Motions and Chaos in Nonlinear Systems pp 173–202Cite as

Hidden dimensions in an Hamiltonian system on networks

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Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 15))

Abstract

In this chapter we show how the topology of a particular network model influences the thermodynamic behavior of a dynamical system defined on it, namely, the Hamiltonian XY rotor model. More specifically, following an introduction, we first consider the regular networks described in De Nigris and Leoncini (EPL, 101(1):10002, 2013). We show analytically that by reducing the degree to only two links per node, long-range order is still present, provided that one of the links connects two nodes that are “far enough,” that is, beyond the \(\sqrt{ N}\) threshold. The results are then confirmed numerically. Given these findings, we return to the network topology. We introduce the notion of an effective (fractal-like) network dimension and devise a method of building a class of networks (lace networks) with a given underlying dimension, \(d \in [1,\:+\infty [\), that can be tuned either by using a rewiring probability or by changing with the considered system’s size. Our findings point to d c  = 2 as the critical dimension between networks displaying long-range order and those where it is absent. We show as well that the critical so-called chaotic states arising in De Nigris and Leoncini (EPL, 101(1):10002, 2013) can be recovered by building lace networks with d = 2. We have therefore devised a more generic and general way to construct networks capable of sustaining infinite susceptibility over a finite range of energies, i.e., possibly robust to eventual external quantitative thermal variations.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Namur Center for Complex Systems-naXys, University of Namur, 8 rempart de la Vierge, 5000, Namur, Belgium

    Sarah de Nigris

  2. Centre de Physique Théorique, Luminy Case 907, 13288, Marseille Cedex 9, France

    Xavier Leoncini

Authors
  1. Sarah de Nigris
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  2. Xavier Leoncini
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Correspondence to Sarah de Nigris .

Editor information

Editors and Affiliations

  1. San Luis Potosi University, San Luis Potosi, Mexico

    Valentin Afraimovich

  2. Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    José António Tenreiro Machado

  3. School of Energy&PowerEngineering, Xi’an Jiaotong University, Shaanxi Province, Shaanxi, China

    Jiazhong Zhang

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de Nigris, S., Leoncini, X. (2016). Hidden dimensions in an Hamiltonian system on networks. In: Afraimovich, V., Machado, J., Zhang, J. (eds) Complex Motions and Chaos in Nonlinear Systems. Nonlinear Systems and Complexity, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-28764-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-28764-5_6

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  • Online ISBN: 978-3-319-28764-5

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