Abstract
Complex phenomena are observed in various situations. These complex phenomena are produced from deterministic dynamical systems or stochastic systems. Then, it is an important issue to clarify what is a source of the complex phenomena and to analyze what kind of response will emerge. Then, in this paper, we analyze deterministic chaos from a new aspect. The analysis method is based on the idea that attractors of nonlinear dynamical systems and networks are characterized by a two-dimensional matrix: a recurrence plot and an adjacent matrix. Then, we transformed the attractors to the networks, and evaluated the clustering coefficients and the characteristic path length to the networks. As a result, the networks constructed from the chaotic systems show a small world property.
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References
e.q. Ruelle, D.: Chaotic evolution and strange attractors. Cambridge University Press, Cambridge (1989)
Thiel, M., Romano, M.C., Kurths, J., Rolfs, M., Kliegl, R.: Twin surrogates to test for complex synchronisation. Europhysics Letters 74, 535–541 (2006)
Thiel, M., Romano, M.C.: Estimation of dynamical invariants without embedding by recurrence plots. Chaos: An Interdisciplinary Journal of Nonlinear Science 14(2), 234–243 (2004)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Costa, L.D.F., Rodrigues, F.A., Travieso, G., Villas Boas, P.R.: Characterization of complex networks: A survey of measurements. Advances in Physics 56, 167–242 (2007)
Zhang, J., Small, M.: Complex network from pseudoperiodic time series: Topology versus dynamics. Physical Review Letters 96(238701) (2006)
Packard, N.H., Crutchfield, J.P., Farmar, J.D., Shaw, R.S.: Geometry from a time series. Physical Review Letters 45(9), 712–716 (1980)
Rossler, O.E.: An equation for continuous chaos. Physics Letters A 57(5), 397–398 (1976)
Lorenz, E.N.: Deterministic nonperiodic flow. Journal of The Atmospheric Sciences 20, 131–141 (1963)
Weigend, A.S., Gershenfeld, N.A.: Time series prediction: Forecasting the future and understanding the past. Santa Fe Institute (1992)
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Shimada, Y., Kimura, T., Ikeguchi, T. (2008). Analysis of Chaotic Dynamics Using Measures of the Complex Network Theory. In: Kůrková, V., Neruda, R., Koutník, J. (eds) Artificial Neural Networks - ICANN 2008. ICANN 2008. Lecture Notes in Computer Science, vol 5163. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87536-9_7
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DOI: https://doi.org/10.1007/978-3-540-87536-9_7
Publisher Name: Springer, Berlin, Heidelberg
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