×

Disentangling regular and chaotic motion in the standard map using complex network analysis of recurrences in phase space. (English) Zbl 1390.37136

Summary: Recurrence in the phase space of complex systems is a well-studied phenomenon, which has provided deep insights into the nonlinear dynamics of such systems. For dissipative systems, characteristics based on recurrence plots have recently attracted much interest for discriminating qualitatively different types of dynamics in terms of measures of complexity, dynamical invariants, or even structural characteristics of the underlying attractor’s geometry in phase space. Here, we demonstrate that the latter approach also provides a corresponding distinction between different co-existing dynamical regimes of the standard map, a paradigmatic example of a low-dimensional conservative system. Specifically, we show that the recently developed approach of recurrence network analysis provides potentially useful geometric characteristics distinguishing between regular and chaotic orbits. We find that chaotic orbits in an intermittent laminar phase (commonly referred to as sticky orbits) have a distinct geometric structure possibly differing in a subtle way from those of regular orbits, which is highlighted by different recurrence network properties obtained from relatively short time series. Thus, this approach can help discriminating regular orbits from laminar phases of chaotic ones, which presents a persistent challenge to many existing chaos detection techniques.{
©2016 American Institute of Physics}

MSC:

37M10 Time series analysis of dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Software:

K2
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Poincaré, H., Sur la problème des trois corps et les équations de la dynamique, Acta Math., 13, A3-A270 (1890) · JFM 22.0907.01
[2] Eckmann, J.-P.; Kamphorst, S. O.; Ruelle, D., Recurrence plots of dynamical systems, Europhys. Lett., 4, 973-977 (1987) · doi:10.1209/0295-5075/4/9/004
[3] Marwan, N.; Romano, M. C.; Thiel, M.; Kurths, J., Recurrence plots for the analysis of complex systems, Phys. Rep., 438, 237-329 (2007) · doi:10.1016/j.physrep.2006.11.001
[4] Faure, P.; Korn, H., A new method to estimate the Kolmogorov entropy from recurrence plots: Its application to neuronal signals, Physica D, 122, 265-279 (1998) · doi:10.1016/S0167-2789(98)00177-8
[5] Thiel, M.; Romano, M.; Read, P.; Kurths, J., Estimation of dynamical invariants without embedding by recurrence plots, Chaos, 14, 234-243 (2004) · Zbl 1080.37091 · doi:10.1063/1.1667633
[6] Zbilut, J. P.; Webber, C. L. Jr., Embeddings and delays as derived from quantification of recurrence plots, Phys. Lett. A, 171, 199-203 (1992) · doi:10.1016/0375-9601(92)90426-M
[7] Marwan, N.; Wessel, N.; Meyerfeldt, U.; Schirdewan, A.; Kurths, J., Recurrence plot based measures of complexity and its application to heart rate variability data, Phys. Rev. E, 66, 026702 (2002) · doi:10.1103/PhysRevE.66.026702
[8] Marwan, N.; Donges, J. F.; Zou, Y.; Donner, R. V.; Kurths, J., Complex network approach for recurrence analysis of time series, Phys. Lett. A, 373, 4246-4254 (2009) · Zbl 1234.05214 · doi:10.1016/j.physleta.2009.09.042
[9] Donner, R. V.; Zou, Y.; Donges, J. F.; Marwan, N.; Kurths, J., Recurrence networks—A novel paradigm for nonlinear time series analysis, New J. Phys., 12, 033025 (2010) · Zbl 1360.90045 · doi:10.1088/1367-2630/12/3/033025
[10] Donner, R. V.; Small, M.; Donges, J. F.; Marwan, N.; Zou, Y.; Xiang, R.; Kurths, J., Recurrence-based time series analysis by means of complex network methods, Int. J. Bifurcation Chaos, 21, 1019-1046 (2011) · Zbl 1247.37086 · doi:10.1142/S0218127411029021
[11] Karney, C., Long-time correlations in the stochastic regime, Physica D, 8, 360-380 (1983) · doi:10.1016/0167-2789(83)90232-4
[12] Lichtenberg, A.; Lieberman, M., Regular and Chaotic Dynamics (1992) · Zbl 0748.70001
[13] Meiss, J., Symplectic maps, variational principles, and transport, Rev. Mod. Phys., 64, 795-848 (1992) · Zbl 1160.37302 · doi:10.1103/RevModPhys.64.795
[14] Bountis, T. C.; Skokos, C., Complex Hamiltonian Dynamics (2012) · Zbl 1248.37001
[15] Afraimovich, V.; Zaslavsky, G.; Benkadda, S.; Zaslavsky, G., Sticky orbits of chaotic Hamiltonian dynamics, Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas, 59-82 (1998) · Zbl 0929.37020
[16] Zaslavsky, G., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[17] Ott, E., Chaos in Dynamical Systems (1993) · Zbl 0792.58014
[18] Afraimovich, V.; Zaslavsky, G. M., Spacetime complexity in Hamiltonian dynamics, Chaos, 13, 519-532 (2003) · Zbl 1080.37581 · doi:10.1063/1.1566171
[19] Kandrup, H.; Siopis, C.; Contopoulos, G.; Dvorak, R., Diffusion and scaling in escapes from two degrees of freedom Hamiltonian systems, Chaos, 9, 381-392 (1999) · Zbl 0982.37063 · doi:10.1063/1.166415
[20] Contopoulos, G.; Voglis, N.; Dvorak, R., Transition spectra of dynamical systems, Cel. Mech. Dyn. Astron., 67, 293-317 (1997) · Zbl 0905.58039 · doi:10.1023/A:1008275829979
[21] Zou, Y.; Pazó, D.; Thiel, M.; Romano, M.; Kurths, J., Distinguishing quasiperiodic dynamics from chaos in short-time series, Phys. Rev. E, 76, 016210 (2007) · doi:10.1103/PhysRevE.76.016210
[22] Zou, Y.; Thiel, M.; Romano, M.; Kurths, J., Characterization of stickiness by means of recurrence, Chaos, 17, 043101 (2007) · Zbl 1163.37393 · doi:10.1063/1.2785159
[23] Trulla, L. L.; Giuliani, A.; Zbilut, J. P.; Webber, C. L. Jr., Recurrence quantification analysis of the logistic equation with transients, Phys. Lett. A, 223, 255-260 (1996) · Zbl 1037.37507 · doi:10.1016/S0375-9601(96)00741-4
[24] Thiel, M.; Romano, M.; Kurths, J., Analytical description of recurrence plots of white noise and chaotic processes, Appl. Nonlinear Dyn., 11, 20-29 (2003) · Zbl 1116.37321
[25] Altmann, E. G.; Kantz, H., Recurrence time analysis, long-term correlations, and extreme events, Phys. Rev. E, 71, 56106 (2005) · doi:10.1103/PhysRevE.71.056106
[26] Slater, N., Gaps and steps for the sequence \(<mml:math display=''inline`` overflow=''scroll``>\) mod 1, Proc. Cambridge Philos. Soc., 63, 1115-1123 (1967) · Zbl 0178.04703 · doi:10.1017/S0305004100042195
[27] Altmann, E. G.; Cristadoro, G.; Pazó, D., Nontwist non-Hamiltonian systems, Phys. Rev. E, 73, 056201 (2006) · doi:10.1103/PhysRevE.73.056201
[28] Baptista, M.; Kraut, S.; Grebogi, C., Poincaré recurrence and measure of hyperbolic and nonhyperbolic chaotic attractors, Phys. Rev. Lett., 95, 094101 (2005) · doi:10.1103/PhysRevLett.95.094101
[29] Faranda, D.; Lucarini, V.; Turchetti, G.; Vaienti, S., Generalized extreme value distribution parameters as dynamical indicators of stability, Int. J. Bifurcation Chaos, 22, 1250276 (2012) · Zbl 1258.37030 · doi:10.1142/S0218127412502768
[30] Chirikov, B. V.; Shepelyansky, D. L., Correlation properties of dynamical chaos in Hamiltonian systems, Physica D, 13, 395-400 (1984) · Zbl 0588.58039 · doi:10.1016/0167-2789(84)90140-4
[31] Zou, Y.; Donner, R. V.; Donges, J. F.; Marwan, N.; Kurths, J., Identifying shrimps in continuous dynamical systems using recurrence-based methods, Chaos, 20, 043130 (2010) · doi:10.1063/1.3523304
[32] Donner, R. V.; Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J., The geometry of chaotic dynamics—A complex network perspective, Eur. Phys. J. B, 84, 653-672 (2011) · Zbl 1515.37092 · doi:10.1140/epjb/e2011-10899-1
[33] Donges, J. F.; Donner, R. V.; Rehfeld, K.; Marwan, N.; Trauth, M. H.; Kurths, J., Identification of dynamical transitions in marine palaeoclimate records by recurrence network analysis, Nonlinear Proc. Geophys., 18, 545-562 (2011) · doi:10.5194/npg-18-545-2011
[34] Donges, J. F.; Donner, R. V.; Trauth, M. H.; Marwan, N.; Schellnhuber, H. J.; Kurths, J., Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution, Proc. Natl. Acad. Sci. U.S.A., 108, 20422-20427 (2011) · doi:10.1073/pnas.1117052108
[35] Donges, J. F.; Heitzig, J.; Donner, R. V.; Kurths, J., Analytical framework for recurrence network analysis of time series, Phys. Rev. E, 85, 046105 (2012) · doi:10.1103/PhysRevE.85.046105
[36] Tsiganis, K.; Anastasiadis, A.; Varvoglis, H., Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system, Chaos, Solitons Fractals, 11, 2281-2292 (2000) · Zbl 0957.37066 · doi:10.1016/S0960-0779(99)00147-2
[37] Donner, R. V.; Zou, Y.; Donges, J. F.; Marwan, N.; Kurths, J., Ambiguities in recurrence-based complex network representations of time series, Phys. Rev. E, 81, 015101(R) (2010) · doi:10.1103/PhysRevE.81.015101
[38] Freistetter, F., Fractal dimensions as chaos indicators, Cel. Mech. Dyn. Astron., 78, 211-225 (2000) · Zbl 1083.37511 · doi:10.1023/A:1011157505026
[39] Zaslavsky, G., Physics of Chaos in Hamiltonian Systems (1998) · Zbl 0913.58041
[40] Meiss, J. D., Class renormalization: Islands around islands, Phys. Rev. A, 34, 2375-2383 (1986) · doi:10.1103/PhysRevA.34.2375
[41] Hénon, M.; Heiles, C., The applicability of the third integral of motion: some numerical experiments, Astronom. J., 69, 73-79 (1964) · doi:10.1086/109234
[42] Gottwald, G. A.; Skokos, C., Preface to the focus issue: Chaos detection methods and predictability, Chaos, 24, 024201 (2014) · doi:10.1063/1.4884603
[43] Asghari, N.; Broeg, C.; Carone, L.; Casas-Miranda, R.; Palacio, J. C. C.; Csillik, I.; Dvorak, R.; Freistetter, F.; Hadjivantsides, G.; Hussmann, H.; Khramova, A.; Khristoforova, M.; Khromova, I.; Kitiashivilli, I.; Kozlowski, S.; Laakso, T.; Laczkowski, T.; Lytvinenko, D.; Miloni, O.; Morishima, R.; Moro-Martin, A.; Paksyutov, V.; Pal, A.; Patidar, V.; Pecnik, B.; Peles, O.; Pyo, J.; Quinn, T.; Rodriguez, A.; Romano, C.; Saikia, E.; Stadel, J.; Thiel, M.; Todorovic, N.; Veras, D.; Neto, E. V.; Vilagi, J.; von Bloh, W.; Zechner, R.; Zhuchkova, E., Stability of terrestrial planets in the habitable zone of GI77A, HD72659, GI614, 47Uma and HD4208, Astron. Astrophys., 426, 353-365 (2004) · doi:10.1051/0004-6361:20040390
[44] Zou, Y.; Thiel, M.; Romano, M. C.; Bi, Q.; Kurths, J., Shrimp structure and associated dynamics in parametrically excited oscillators, Int. J. Bifurcation Chaos, 16, 3567-3579 (2006) · Zbl 1117.37027 · doi:10.1142/S0218127406016987
[45] Zhang, J.; Small, M., Complex network from pseudoperiodic time series: Topology versus dynamics, Phys. Rev. Lett., 96, 238701 (2006) · doi:10.1103/PhysRevLett.96.238701
[46] Xu, X.; Zhang, J.; Small, M., Superfamily phenomena and motifs of networks induced from time series, Proc. Natl. Acad. Sci. U.S.A., 105, 19601-19605 (2008) · Zbl 1202.37118 · doi:10.1073/pnas.0806082105
[47] Lacasa, L.; Luque, B.; Ballesteros, F.; Luque, J.; Nuno, J. C., From time series to complex networks: The visibility graph, Proc. Natl. Acad. Sci. U.S.A., 105, 4972-4975 (2008) · Zbl 1205.05162 · doi:10.1073/pnas.0709247105
[48] Facchini, A.; Kantz, H., Curved structures in recurrence plots: The role of the sampling time, Phys. Rev. E, 75, 036215 (2007) · doi:10.1103/PhysRevE.75.036215
[49] Newman, M. E. J., The structure and function of complex networks, SIAM Rev., 45, 167-256 (2003) · Zbl 1029.68010 · doi:10.1137/S003614450342480
[50] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: Structure and dynamics, Phys. Rep., 424, 175-308 (2006) · Zbl 1371.82002 · doi:10.1016/j.physrep.2005.10.009
[51] Watts, D. J.; Strogatz, S. H., Collective dynamics of “small-world” networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139 · doi:10.1038/30918
[52] Barrat, A.; Weigt, M., On the properties of small-world network models, Eur. Phys. J. B, 13, 547-560 (2000) · doi:10.1007/s100510050067
[53] Newman, M. E. J., Scientific collaboration networks. I. Network construction and fundamental results, Phys. Rev. E, 64, 016131 (2001) · doi:10.1103/PhysRevE.64.016131
[54] Gottwald, G. A.; Melbourne, I., A new test for chaos in deterministic systems, Proc. R. Soc. London A, 460, 603-611 (2004) · Zbl 1042.37060 · doi:10.1098/rspa.2003.1183
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.