| In the last few years, several approaches have been proposed for studying properties of dynamical systems represented by time series by means of complex network methods [1,2]. Recently, two main concepts have attracted particular interest: recurrence networks and visibility graphs. Here, I will present a thorough review of the state of the art of both approaches, with a special emphasis on what kind of information about a dynamical system can be inferred from its adjoint network representations. On the one hand, recurrence networks are based on the mutual proximity of sampled state vectors in the (possibly reconstructed) phase space of a dynamical system under study. Their local as well as global properties have been proven to characterize important structural aspects of the underlying attractors (e.g., scale-free degree distributions highlight power-law singularities of the invariant densities, clustering/transitivity properties are related with a novel notion of fractal dimension of the system [3],...). Successful and relevant real-world applications particularly include the identification of dynamical transitions in observational time series [4,5]. On the other hand, visibility graphs and related concepts traditionally characterize stochastic properties of a system such as its Hurst exponent or a possible time-reversal asymmetry. The potentials and possible methodological problems of both approaches are highlighted and illustrated based on some real-world observational time series [4,5,6]. [1] R.V. Donner et al., New J. Phys., 12, 033025, 2010 [2] R.V. Donner et al., Int. J. Bifurcation Chaos, 21, 1019, 2011 [3] R.V. Donner et al., Eur. Phys. J. B, 84, 653, 2011 [4] J.F. Donges et al., PNAS, 108, 20422, 2011 [5] J.F. Donges et al., Nonlin. Proc. Geophys., 18, 545, 2011 [6] R.V. Donner and J.F. Donges, Acta Geophys., in press |
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