{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:44:05Z","timestamp":1760233445164,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T00:00:00Z","timestamp":1610409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006595","name":"Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii","doi-asserted-by":"publisher","award":["PN-III-P1-1.1-TE-2016-1457","PN-III-P1-1.1-PD-2019-0742"],"award-info":[{"award-number":["PN-III-P1-1.1-TE-2016-1457","PN-III-P1-1.1-PD-2019-0742"]}],"id":[{"id":"10.13039\/501100006595","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.<\/jats:p>","DOI":"10.3390\/e23010103","type":"journal-article","created":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T14:28:53Z","timestamp":1610461733000},"page":"103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4887-7859","authenticated-orcid":false,"given":"Bulcs\u00fa","family":"S\u00e1ndor","sequence":"first","affiliation":[{"name":"Department of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bence","family":"Schneider","sequence":"additional","affiliation":[{"name":"Department of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7356-1683","authenticated-orcid":false,"given":"Zsolt I.","family":"L\u00e1z\u00e1r","sequence":"additional","affiliation":[{"name":"Department of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4428-9953","authenticated-orcid":false,"given":"M\u00e1ria","family":"Ercsey-Ravasz","sequence":"additional","affiliation":[{"name":"Department of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania"},{"name":"Network Science Lab, Transylvanian Institute of Neuroscience, 400157 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Newman, M. 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A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.cnsns.2016.04.028","article-title":"Characterization of chaotic attractors under noise: A recurrence network perspective","volume":"41","author":"Jacob","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jacob, R., Harikrishnan, K.P., Misra, R., and Ambika, G. (2016). Uniform framework for the recurrence-network analysis of chaotic time series. Phys. Rev. E, 93.","DOI":"10.1103\/PhysRevE.93.012202"},{"key":"ref_7","first-page":"20110623","article-title":"Classification of cardiovascular time series based on different coupling structures using recurrence networks analysis","volume":"371","author":"Gapelyuk","year":"2013","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.icheatmasstransfer.2015.02.014","article-title":"Detection of two-phase flow patterns using the recurrence network analysis of pressure drop fluctuations","volume":"64","author":"Mosdorf","year":"2015","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"30007","DOI":"10.1209\/0295-5075\/102\/30007","article-title":"Geometric signature of complex synchronisation scenarios","volume":"102","author":"Feldhoff","year":"2013","journal-title":"EPL Europhys. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"de Berg, M., van Kreveld, M., Overmars, M., and Schwarzkopf, O.C. (2000). Computational Geometry, Springer.","DOI":"10.1007\/978-3-662-04245-8"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"589","DOI":"10.2478\/s11600-012-0032-x","article-title":"Visibility graph analysis of geophysical time series: Potentials and possible pitfalls","volume":"60","author":"Donner","year":"2012","journal-title":"Acta Geophys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6571","DOI":"10.1016\/j.physa.2013.08.078","article-title":"Investigating the time dynamics of seismicity by using the visibility graph approach: Application to seismicity of Mexican subduction zone","volume":"392","author":"Telesca","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gao, Z.K., Cai, Q., Yang, Y.X., Dang, W.D., and Zhang, S.S. (2016). Multiscale limited penetrable horizontal visibility graph for analyzing nonlinear time series. Sci. Rep., 6.","DOI":"10.1038\/srep35622"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4720","DOI":"10.1016\/j.physa.2012.04.025","article-title":"Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder","volume":"391","author":"Ahmadlou","year":"2012","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1140\/epjst\/e2008-00836-2","article-title":"Symbolic recurrence plots: A new quantitative framework for performance analysis of manufacturing networks","volume":"164","author":"Donner","year":"2008","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Small, M. (2013, January 19\u201323). Complex networks from time series: Capturing dynamics. Proceedings of the 2013 IEEE International Symposium on Circuits and Systems (ISCAS2013), Beijing, China.","DOI":"10.1109\/ISCAS.2013.6572389"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Deritei, D., Aird, W.C., Ercsey-Ravasz, M., and Regan, E.R. (2016). Principles of dynamical modularity in biological regulatory networks. Sci. Rep., 6.","DOI":"10.1038\/srep21957"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"053101","DOI":"10.1063\/1.4919075","article-title":"Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems","volume":"25","author":"McCullough","year":"2015","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"073114","DOI":"10.1063\/1.4959537","article-title":"Using ordinal partition transition networks to analyze ECG data","volume":"26","author":"Kulp","year":"2016","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Antoniades, I.P., Stavrinides, S.G., Hanias, M.P., and Magafas, L. (2020). Complex Network Time Series Analysis of a Macroeconomic Model. Chaos and Complex Systems, Springer International Publishing.","DOI":"10.1007\/978-3-030-35441-1_13"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"043111","DOI":"10.1063\/1.5086527","article-title":"Ordinal partition transition network based complexity measures for inferring coupling direction and delay from time series","volume":"29","author":"Ruan","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gagniuc, P.A. (2017). Markov Chains: From Theory to Implementation and Experimentation, John Wiley & Sons.","DOI":"10.1002\/9781119387596"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/0-387-21525-5_1","article-title":"Markov Chains","volume":"51","author":"Asmussen","year":"2003","journal-title":"Appl. Probab. Queues. Stoch. Model. Appl. Probab."},{"key":"ref_24","first-page":"768","article-title":"On the Notion of Entropy of a Dynamical System","volume":"124","author":"Sinai","year":"1959","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pikovsky, A., and Politi, A. (2016). Lyapunov Exponents, a Tool to Explore Complex Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139343473"},{"key":"ref_26","unstructured":"Kantz, H., and Schreiber, T. (2010). Nonlinear Time Series Analysis, Cambridge University Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF01608556","article-title":"A two-dimensional mapping with a strange attractor","volume":"50","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.neuron.2013.07.036","article-title":"A Predictive Network Model of Cerebral Cortical Connectivity Based on a Distance Rule","volume":"80","author":"Markov","year":"2013","journal-title":"Neuron"},{"key":"ref_30","unstructured":"Gantmakher, F., Hirsch, K., Society, A.M., and Collection, K.M.R. (1959). The Theory of Matrices, Chelsea Publishing Company."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1038\/s41598-017-01083-x","article-title":"How to test for partially predictable chaos","volume":"7","author":"Wernecke","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"T\u00e9l, T., and Gruiz, M. (2006). Chaotic Dynamics: An Introduction Based on Classical Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511803277"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/103\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:10:11Z","timestamp":1760159411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,12]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010103"],"URL":"https:\/\/doi.org\/10.3390\/e23010103","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,1,12]]}}}