{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T10:49:24Z","timestamp":1768733364223,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,7,27]],"date-time":"2018-07-27T00:00:00Z","timestamp":1532649600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["1174715"],"award-info":[{"award-number":["1174715"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Startup Foundation for Doctoral Research in Hunan University of Arts and Science","award":["E07016048"],"award-info":[{"award-number":["E07016048"]}]},{"name":"the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province","award":["16KJB120004"],"award-info":[{"award-number":["16KJB120004"]}]},{"name":"the Startup Foundation for Introducing Talent of NUIST","award":["2016205"],"award-info":[{"award-number":["2016205"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel\u2013Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems.<\/jats:p>","DOI":"10.3390\/e20080556","type":"journal-article","created":{"date-parts":[[2018,7,27]],"date-time":"2018-07-27T12:20:03Z","timestamp":1532694003000},"page":"556","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5190-4841","authenticated-orcid":false,"given":"Shaobo","family":"He","sequence":"first","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunbiao","family":"Li","sequence":"additional","affiliation":[{"name":"Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science &amp; Technology, Nanjing 210044, China"},{"name":"Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science &amp; Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kehui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sajad","family":"Jafari","sequence":"additional","affiliation":[{"name":"Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1142\/S0218127408021233","article-title":"Complex dynamics in multistable systems","volume":"18","author":"Feudel","year":"2008","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s40435-017-0332-8","article-title":"Multistability and hidden chaotic attractors in a new simple 4-D chaotic system with chaotic 2-torus behaviour","volume":"6","author":"Singh","year":"2017","journal-title":"Int. J. Dyn. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1140\/epjst\/e2015-02475-x","article-title":"Multistability in symmetric chaotic systems","volume":"224","author":"Li","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1007\/s11071-016-3310-3","article-title":"Controlling multistability in a vibro-impact capsule system","volume":"88","author":"Liu","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.physd.2015.05.005","article-title":"Multistability and hidden attractors in a relay system with hysteresis","volume":"306","author":"Zhusubaliyev","year":"2015","journal-title":"Phys. D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"043111","DOI":"10.1063\/1.4946813","article-title":"Multistability in Chua\u2019s circuit with two stable node-foci","volume":"26","author":"Bao","year":"2016","journal-title":"Chaos"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3237","DOI":"10.1016\/j.cnsns.2010.12.015","article-title":"Multi-stable chaotic attractors in generalized synchronization","volume":"16","author":"Hu","year":"2011","journal-title":"Commun. Nonl. Sci. Num. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1007\/s11071-017-3752-2","article-title":"Multistability analysis, circuit implementations and application in image encryption of a novel memristive chaotic circuit","volume":"90","author":"Peng","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1137\/050632440","article-title":"Multistability in recurrent neural networks","volume":"66","author":"Shih","year":"2006","journal-title":"SIAM J. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3000","DOI":"10.1016\/j.ijleo.2015.12.089","article-title":"Research on a new 3D autonomous chaotic system with coexisting attractors","volume":"127","author":"Lai","year":"2016","journal-title":"Optik"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1750160","DOI":"10.1142\/S0218127417501607","article-title":"Infinite multistability in a self-reproducing chaotic system","volume":"27","author":"Li","year":"2017","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2629","DOI":"10.1007\/s11071-017-3612-0","article-title":"An infinite 2-D lattice of strange attractors","volume":"89","author":"Li","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/S0020-7225(02)00211-2","article-title":"Kolmogorov complexity and chaotic phenomena","volume":"41","author":"Kreinovich","year":"2003","journal-title":"Int. J. Eng. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2564","DOI":"10.1016\/j.physa.2011.12.042","article-title":"Sampling period, statistical complexity, and chaotic attractors","volume":"391","author":"Micco","year":"2012","journal-title":"Phys. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s12043-015-0938-3","article-title":"Classification of periodic, chaotic and random sequences using approximate entropy and Lempel\u2013Ziv complexity measures","volume":"84","author":"Balasubramanian","year":"2015","journal-title":"Pramana"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1714","DOI":"10.1134\/S1063784217110202","article-title":"Application of permutation entropy method in the analysis of chaotic, noisy, and chaotic noisy series","volume":"62","author":"Makark","year":"2017","journal-title":"Tech. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8299","DOI":"10.3390\/e17127882","article-title":"Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System","volume":"17","author":"He","year":"2015","journal-title":"Entropy"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1002\/cplx.21629","article-title":"Modified multiscale permutation entropy algorithm and its application for multiscroll chaotic systems","volume":"21","author":"He","year":"2016","journal-title":"Complexity"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.physa.2015.07.030","article-title":"Can complexity decrease in congestive heart failure?","volume":"439","author":"Mukherjee","year":"2015","journal-title":"Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"061918","DOI":"10.1103\/PhysRevE.84.061918","article-title":"Multivariate multiscale entropy: A tool for complexity analysis of multichannel data","volume":"84","author":"Ahmed","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/B978-0-12-381270-4.00013-5","article-title":"Multivariate neighborhood sample entropy: A method for data reduction and prediction of complex data","volume":"487","author":"Richman","year":"2011","journal-title":"Meth. Enzymol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1016\/j.physa.2016.06.012","article-title":"Multivariate permutation entropy and its application for complexity analysis of chaotic systems","volume":"461","author":"He","year":"2016","journal-title":"Phys. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"021906","DOI":"10.1103\/PhysRevE.71.021906","article-title":"Multiscale entropy analysis of biological signals","volume":"71","author":"Costa","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3110","DOI":"10.3390\/e17053110","article-title":"The Multiscale entropy algorithm and its variants: A review","volume":"17","author":"Humeauheurtier","year":"2015","journal-title":"Entropy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1007\/s40846-015-0035-6","article-title":"Wavelet-Based Multiscale Sample Entropy and Chaotic Features for Congestive Heart Failure Recognition Using Heart Rate Variability","volume":"35","author":"Yu","year":"2015","journal-title":"J. Med. Biol. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TIT.1976.1055501","article-title":"On the complexity of finite sequences","volume":"22","author":"Lempel","year":"1976","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1140\/epjb\/e2014-41018-5","article-title":"Mixing Bandt-Pompe and Lempel\u2013Ziv approaches: Another way to analyze the complexity of continuous-state sequences","volume":"87","author":"Zozor","year":"2014","journal-title":"Eur. Phys. J. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.jsv.2008.07.011","article-title":"Fault severity assessment for rolling element bearings using the Lempel\u2013Ziv complexity and continuous wavelet transform","volume":"320","author":"Hong","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1650233","DOI":"10.1142\/S0218127416502333","article-title":"Crisis in amplitude control hides in multistability","volume":"26","author":"Li","year":"2017","journal-title":"Inter. J. Bifur. Chaos"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1007\/s11071-016-3118-1","article-title":"Constructing chaotic systems with conditional symmetry","volume":"87","author":"Li","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s11071-015-2364-y","article-title":"Dynamical analysis of a simple autonomous jerk system with multiple attractors","volume":"83","author":"Kengne","year":"2016","journal-title":"Nonlinear Dyn."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/556\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:14:40Z","timestamp":1760195680000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/556"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,27]]},"references-count":32,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["e20080556"],"URL":"https:\/\/doi.org\/10.3390\/e20080556","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,27]]}}}