NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-21-1113-2014 Long-term changes in the north–south asymmetry of solar activity: a nonlinear dynamics characterization using visibility graphs Zou Y. 1 2 3 Donner R. V. https://orcid.org/0000-0001-7023-6375 3 4 Marwan N. https://orcid.org/0000-0003-1437-7039 3 Small M. 5 Kurths J. 3 6 7 Department of Physics, East China Normal University, 200062 Shanghai, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany Max Planck Institute for Biogeochemistry, Hans-Knöll-Str. 10, 07745 Jena, Germany School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen, AB243UE, UK 24 11 2014 21 6 1113 1126 Copyright: © 2014 Y. Zou et al. 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/21/1113/2014/npg-21-1113-2014.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/21/1113/2014/npg-21-1113-2014.pdf

Solar activity is characterized by complex dynamics superimposed onto an almost periodic, approximately 11-year cycle. One of its main features is the presence of a marked, time-varying hemispheric asymmetry, the deeper reasons for which have not yet been completely uncovered. Traditionally, this asymmetry has been studied by considering amplitude and phase differences. Here, we use visibility graphs, a novel tool of nonlinear time series analysis, to obtain complementary information on hemispheric asymmetries in dynamical properties. Our analysis provides deep insights into the potential and limitations of this method, revealing a complex interplay between factors relating to statistical and dynamical properties, i.e., effects due to the probability distribution and the regularity of observed fluctuations. We demonstrate that temporal changes in the hemispheric predominance of the graph properties lag those directly associated with the total hemispheric sunspot areas. Our findings open a new dynamical perspective on studying the north–south sunspot asymmetry, which is to be further explored in future work.

 National Natural Science Foundation of China 11305062, 11135001, 11075056 20130076120003 References Agnihotri, R., Dutta, K., Bhushan, R., and Somayajulu, B.: Evidence for solar forcing on the Indian monsoon during the last millennium, Earth Planet. Sc. Lett., 198, 521–527, <a href="http://dx.doi.org/10.1016/S0012-821X(02)00530-7">https://doi.org/10.1016/S0012-821X(02)00530-7</a>, 2002. Ahmadlou, M. and Adeli, H.: Visibility graph similarity: A new measure of generalized synchronization in coupled dynamic systems, Physica D, 241, 326–332, <a href="http://dx.doi.org/10.1016/j.physd.2011.09.008">https://doi.org/10.1016/j.physd.2011.09.008</a>, 2012. Bard, E. and Frank, M.: Climate change and solar variability: What's new under the Sun?, Earth Planet. Sc. Lett., 248, 480–493, <a href="http://dx.doi.org/10.1016/j.epsl.2006.06.016">https://doi.org/10.1016/j.epsl.2006.06.016</a>, 2006. Barrat, A. and Weigt, M.: On the properties of small-world network models, Eur. Phys. J. B, 13, 547–560, <a href="http://dx.doi.org/10.1007/s100510050067">https://doi.org/10.1007/s100510050067</a>, 2000. Bond, G., Kromer, B., Beer, J., Muscheler, R., Evans, M. N., Showers, W., Hoffmann, S., Lotti-Bond, R., Hajdas, I., and Bonani, G.: Persistent solar influence on North Atlantic climate during the Holocene, Science, 294, 2130–2136, <a href="http://dx.doi.org/10.1126/science.1065680">https://doi.org/10.1126/science.1065680</a>, 2001. Braj\v sa, R., Wöhl, H., Hanslmeier, A., Verbanac, G., Ru\v zdjak, D., Cliver, E., Svalgaard, L., and Roth, M.: On solar cycle predictions and reconstructions, Astron. Astrophys., 496, 855–861, <a href="http://dx.doi.org/10.1051/0004-6361:200810862">https://doi.org/10.1051/0004-6361:200810862</a>, 2009. Carbonell, M., Oliver, R., and Ballester, J. L.: On the asymmetry of solar activity, Astron. Astrophys., 274, 497–504, 1993. Charbonneau, P.: Dynamo Models of the Solar Cycle, Living Reviews in Solar Physics, 7, 3, <a href="http://dx.doi.org/10.12942/lrsp-2010-3">https://doi.org/10.12942/lrsp-2010-3</a>, 2010. Donges, J. F., Donner, R. V., and Kurths, J.: Testing time series irreversibility using complex network methods, Europhys. Lett., 102, 10004, <a href="http://dx.doi.org/10.1209/0295-5075/102/10004">https://doi.org/10.1209/0295-5075/102/10004</a>, 2013. Donner, R.: Phase Coherence Analysis of Decadal-Scale Sunspot Activity on Both Solar Hemispheres, in: Nonlinear Time Series Analysis in the Geosciences, edited by: Donner, R. V. and Barbosa, S. M., Lect. Notes Earth Sci., Springer, Berlin/Heidelberg, 112, 355–385, <a href="http://dx.doi.org/10.1007/978-3-540-78938-3_16">https://doi.org/10.1007/978-3-540-78938-3_16</a>, 2008. Donner, R. and Thiel, M.: Scale-resolved phase coherence analysis of hemispheric sunspot activity: a new look at the north-south asymmetry, Astron. Astrophys., 475, L33–L36, <a href="http://dx.doi.org/10.1051/0004-6361:20078672">https://doi.org/10.1051/0004-6361:20078672</a>, 2007. Donner, R. V. and Donges, J. F.: Visibility Graph Analysis of Geophysical Time Series: Potentials and Possible Pitfalls, Acta Geophys., 60, 589–623, <a href="http://dx.doi.org/10.2478/s11600-012-0032-x">https://doi.org/10.2478/s11600-012-0032-x</a>, 2012. Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., and Kurths, J.: Recurrence networks – A novel paradigm for nonlinear time series analysis, New J. Phys., 12, 033025, <a href="http://dx.doi.org/10.1088/1367-2630/12/3/033025">https://doi.org/10.1088/1367-2630/12/3/033025</a>, 2010a. Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., and Kurths, J.: Ambiguities in recurrence-based complex network representations of time series, Phys. Rev. E, 81, 015101(R), <a href="http://dx.doi.org/10.1103/PhysRevE.81.015101">https://doi.org/10.1103/PhysRevE.81.015101</a>, 2010b. Donner, R. V., Small, M., Donges, J. F., Marwan, N., Zou, Y., Xiang, R., and Kurths, J.: Recurrence-based time series analysis by means of complex network methods, Int. J. Bifurcat. Chaos, 21, 1019–1046, <a href="http://dx.doi.org/10.1142/S0218127411029021">https://doi.org/10.1142/S0218127411029021</a>, 2011. Elsner, J. B., Jagger, T. H., and Fogarty, E. A.: Visibility network of United States hurricanes, Geophys. Res. Lett., 36, L16702, <a href="http://dx.doi.org/10.1029/2009GL039129">https://doi.org/10.1029/2009GL039129</a>, 2009. Kurths, J. and Ruzmaikin, A. A.: On forecasting the sunspot numbers, Sol. Phys., 126, 407–410, <a href="http://dx.doi.org/10.1007/BF00153060">https://doi.org/10.1007/BF00153060</a>, 1990. Lacasa, L., Luque, B., Ballesteros, F., Luque, J., and Nuño, J. C.: From time series to complex networks: The visibility graph, P. Natl. Acad. Sci. USA, 105, 4972–4975, <a href="http://dx.doi.org/10.1073/pnas.0709247105">https://doi.org/10.1073/pnas.0709247105</a>, 2008. Lacasa, L., Luque, B., Luque, J., and Nuno, J. C.: The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion, Europhys. Lett., 86, 30001, <a href="http://dx.doi.org/10.1209/0295-5075/86/30001">https://doi.org/10.1209/0295-5075/86/30001</a>, 2009. Lacasa, L., Nuñez, A., Roldán, E., Parrondo, J. M. R., and Luque, B.: Time series irreversibility: a visibility graph approach, Eur. Phys. J. B, 85, 217, <a href="http://dx.doi.org/10.1040/epjb/e2012-20809-8">https://doi.org/10.1040/epjb/e2012-20809-8</a>, 2012. Li, K., Gao, P., and Zhan, L.: Synchronization of Sunspot Numbers and Sunspot Areas, Sol. Phys., 255, 289–300, <a href="http://dx.doi.org/10.1007/s11207-009-9328-7">https://doi.org/10.1007/s11207-009-9328-7</a>, 2009. Li, K. J., Gao, P. X., Zhan, L. S., Shi, X. J., and Zhu, W. W.: On the asynchronization of hemispheric high-latitude solar activity, Mon. Not. R. Astron. Soc., 391, L34–L38, <a href="http://dx.doi.org/10.1111/j.1745-3933.2008.00551.x">https://doi.org/10.1111/j.1745-3933.2008.00551.x</a>, 2008. Li, Q.: Periodicity and Hemispheric Phase Relationship in High-Latitude Solar Activity, Sol. Phys., 249, 135–145, <a href="http://dx.doi.org/10.1007/s11207-008-9147-2">https://doi.org/10.1007/s11207-008-9147-2</a>, 2008. Li, Q.-X. and Li, K.-J.: Low Dimensional Chaos from the Group Sunspot Numbers, Chinese J. Astron. Ast., 7, 435, <a href="http://dx.doi.org/10.1088/1009-9271/7/3/14">https://doi.org/10.1088/1009-9271/7/3/14</a>, 2007. Luque, B., Lacasa, L., Ballesteros, F., and Luque, J.: Horizontal visibility graphs: Exact results for random time series, Physical Rev. E, 80, 046103, <a href="http://dx.doi.org/10.1103/PhysRevE.80.046103">https://doi.org/10.1103/PhysRevE.80.046103</a>, 2009. Marwan, N., Donges, J. F., Zou, Y., Donner, R. V., and Kurths, J.: Complex network approach for recurrence analysis of time series, Phys. Lett. A, 373, 4246–4254, <a href="http://dx.doi.org/10.1016/j.physleta.2009.09.042">https://doi.org/10.1016/j.physleta.2009.09.042</a>, 2009. Newton, H. W. and Milsom, A. S.: Note on the observed differences in spottedness of the Sun's northern and southern hemispheres, Mon. Not. R. Astron. Soc., 115, 398–404, 1955. Ni, K.-H., Jiang, Z.-Q., and Zhou, W.-X.: Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks, Phys. Lett. A, 373, 3822–3826, <a href="http://dx.doi.org/10.1016/j.physleta.2009.08.041">https://doi.org/10.1016/j.physleta.2009.08.041</a>, 2009. Nuñez, A. M., Lacasa, L., Gomez, J. P., and Luque, B.: Visibility Algorithms: A Short Review, in: New Frontiers in Graph Theory, edited by: Zhang, Y., InTech, Rijeka, 119–152, <a href="http://dx.doi.org/10.5772/34810">https://doi.org/10.5772/34810</a>, 2012. Petrovay, K.: Solar Cycle Prediction, Living Review in Solar Physics, 7, 6, <a href="http://dx.doi.org/10.12942/lrsp-2010-6">https://doi.org/10.12942/lrsp-2010-6</a>, 2010. Pierini, J. O., Lovallo, M., and Telesca, L.: Visibility graph analysis of wind speed records measured in central Argentina, Physica A, 391, 5041–5048, <a href="http://dx.doi.org/10.1016/j.physa.2012.05.049">https://doi.org/10.1016/j.physa.2012.05.049</a>, 2012. Romano, M. C., Thiel, M., Kurths, J., and von Bloh, W.: Multivariate recurrence plots, Phys. Lett. A, 330, 214–223, <a href="http://dx.doi.org/10.1016/j.physleta.2004.07.066">https://doi.org/10.1016/j.physleta.2004.07.066</a>, 2004. Ruzmaikin, A., Feynman, J., and Robinson, P.: Long-term persistence of solar activity, Solar Physics, 149, 395–403, <a href="http://dx.doi.org/10.1007/978-94-011-0950-5_47">https://doi.org/10.1007/978-94-011-0950-5_47</a>, 1994. Schuster, A.: On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena, Terrestrial Magnetism, 3, 13–41, <a href="http://dx.doi.org/10.1029/TM003i001p00013">https://doi.org/10.1029/TM003i001p00013</a>, 1898. Tang, Q., Liu, J., and Liu, H.: Comparison of different daily streamflow series in US and China, under a viewpoint of complex networks, Mod. Phys. Lett. B, 24, 1541–1547, <a href="http://dx.doi.org/10.1142/S0217984910023335">https://doi.org/10.1142/S0217984910023335</a>, 2010. Telesca, L. and Lovallo, M.: Analysis of seismic sequences by using the method of visibility graph, Europhys. Lett., 97, 50002, <a href="http://dx.doi.org/10.1209/0295-5075/97/50002">https://doi.org/10.1209/0295-5075/97/50002</a>, 2012. Telesca, L., Lovallo, M., and Pierini, J. O.: Visibility graph approach to the analysis of ocean tidal records, Chaos Soliton Fract., 45, 1086–1091, <a href="http://dx.doi.org/10.1016/j.chaos.2012.06.003">https://doi.org/10.1016/j.chaos.2012.06.003</a>, 2012. Voss, H., Kurths, J., and Schwarz, U.: Reconstruction of grand minima of solar activity from Δ<sup>14</sup>C data: Linear and nonlinear signal analysis, J. Geophys. Res., 101, 15637–15643, <a href="http://dx.doi.org/10.1029/96JA00542">https://doi.org/10.1029/96JA00542</a>, 1996. Watts, D. J. and Strogatz, S. H.: Collective dynamics of "small-world" networks, Nature, 393, 440–442, <a href="http://dx.doi.org/10.1038/30918">https://doi.org/10.1038/30918</a>, 1998. Xu, X.-K., Zhang, J., and Small, M.: Superfamily phenomena and motifs of networks induced from time series, P. Natl. Acad. Sci. USA, 105, 19601–19605, <a href="http://dx.doi.org/10.1073/pnas.0806082105">https://doi.org/10.1073/pnas.0806082105</a>, 2008. Yu, Z. G., Anh, V., Eastes, R., and Wang, D.-L.: Multifractal analysis of solar flare indices and their horizontal visibility graphs, Nonlin. Processes Geophys., 19, 657–665, <a href="http://dx.doi.org/10.5194/npg-19-657-2012">https://doi.org/10.5194/npg-19-657-2012</a>, 2012. Yule, G. U.: On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers, Philos. T Roy. Soc. A, 226, 267–298, <a href="http://dx.doi.org/10.1098/rsta.1927.0007">https://doi.org/10.1098/rsta.1927.0007</a>, 1927. Zhang, J. and Small, M.: Complex network from pseudoperiodic time series: Topology versus dynamics, Phys. Rev. Lett., 96, 238701, <a href="http://dx.doi.org/10.1103/PhysRevLett.96.238701">https://doi.org/10.1103/PhysRevLett.96.238701</a>, 2006. Zolotova, N., Ponyavin, D., Marwan, N., and Kurths, J.: Long-term asymmetry in the wings of the butterfly diagram, Astron. Astrophys., 503, 197–201, <a href="http://dx.doi.org/10.1051/0004-6361/200811430">https://doi.org/10.1051/0004-6361/200811430</a>, 2009. Zolotova, N. V. and Ponyavin, D. I.: Phase asynchrony of the north-south sunspot activity, Astron. Astrophys., 449, L1–L4, <a href="http://dx.doi.org/10.1051/0004-6361:200600013">https://doi.org/10.1051/0004-6361:200600013</a>, 2006. Zou, Y., Donner, R. V., Marwan, N., Small, M., and Kurths, J.: <i>Interactive comment on</i> "Long-term changes in the North–South asymmetry of solar activity: a nonlinear dynamics characterization using visibility graphs" <i>by</i> Y. Zou et al., Nonlin. Processes Geophys. Discuss., 1, C323–328, <a href="http://www.nonlin-processes-geophys-discuss.net/1/C323/2014/">http://www.nonlin-processes-geophys-discuss.net/1/C323/2014/</a>, 2014a. Zou, Y., Small, M., Liu, Z., and Kurths, J.: Complex network approach to characterize the statistical features of the sunspot series, New J. Phys., 16, 013051, <a href="http://dx.doi.org/10.1088/1367-2630/16/1/013051">https://doi.org/10.1088/1367-2630/16/1/013051</a>, 2014b.