Abstract
We suggest a network representation of dynamical maps (such as the logistic map) on the unit interval. The unit interval is partitioned into N subintervals, which are associated with ’nodes’ of the network. A link from node i to j is defined as a possible transition of the dynamical map from one interval i, to another j. In this way directed networks more generally allow phasespace representations (i.e. representations of transitions between different phasespace cells), of dynamical maps defined on finite intervals. We compute the diameter of these networks as well as the average path length between any two nodes. We numerically demonstrate that these network characteristics seem to diverge at the (first) zeros of the Lyapunov exponent and could thus provide an alternative measure to detect the ’edge of chaos’ in dynamical systems.
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Robledo, A.: Critical attractors and q-statistics. Europhysics News 6, 214–218 (2005)
Dorogovtsev, S., Mendes, J.F.F.: Evolution of Networks. Oxford University Press, Oxford (2003)
Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002)
Froyland, G., Junge, O., Ochs, G.: Rigorous computation of topological entropy with respect to a finite partition. Physica D 154, 68–84 (2001)
Shreim, A., Grassberger, P., Nadler, W., Samuelsson, B., Socolar, J.E.S., Paczuski, M.: Network analysis of the state space of discrete dynamical systems. cond-mat/0610447 (2006)
Thurner, S.: Nonextensive statistical mechanics and complex scale-free networks. Europhysics News 6, 218–220 (2005)
Borges, E.P., Cajueiro, D.O., Andrade, R.F.S.: Mapping dynamical systems onto complex networks. cond-mat/0610820 (2006)
Park, S.M., Kim, B.J.: Dynamic behaviors in directed networks. Phys. Rev. E 74, 026114 (2006)
Floyd, R.W.: Algorithm 97: Shortest Path. Communications of the ACM 5, 345 (1962)
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Kyriakopoulos, F., Thurner, S. (2007). Directed Network Representation of Discrete Dynamical Maps. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2007. ICCS 2007. Lecture Notes in Computer Science, vol 4488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72586-2_91
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DOI: https://doi.org/10.1007/978-3-540-72586-2_91
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