{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:59:41Z","timestamp":1776779981875,"version":"3.51.2"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"22","license":[{"start":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T00:00:00Z","timestamp":1623801600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T00:00:00Z","timestamp":1623801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2021,11]]},"DOI":"10.1007\/s00521-021-06142-z","type":"journal-article","created":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T19:02:40Z","timestamp":1623870160000},"page":"15191-15203","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network"],"prefix":"10.1007","volume":"33","author":[{"given":"Guo","family":"Luo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qizhi","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,16]]},"reference":[{"key":"6142_CR1","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.1109\/TPWRS.2004.831280","volume":"19","author":"M Karimi-Ghartemani","year":"2004","unstructured":"Karimi-Ghartemani M, Iravani MR (2004) A method for synchronization of power electronic converters in polluted and variable-frequency environments. IEEE Trans Power Syst 19:1263\u20131270","journal-title":"IEEE Trans Power Syst"},{"key":"6142_CR2","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1016\/j.mechatronics.2013.11.010","volume":"24","author":"X Liu","year":"2014","unstructured":"Liu X, Tao R, Tavakoli M (2014) Adaptive control of uncertain nonlinear teleoperation systems. Mechatronics 24:66\u201378","journal-title":"Mechatronics"},{"key":"6142_CR3","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1109\/TBME.2007.899302","volume":"55","author":"O Sayadi","year":"2008","unstructured":"Sayadi O, Shamsollahi MB (2008) Model-based fiducial points extraction for baseline wandered electrocardiograms. IEEE Trans Biomed Eng 55:347\u2013351","journal-title":"IEEE Trans Biomed Eng"},{"key":"6142_CR4","doi-asserted-by":"publisher","first-page":"1832","DOI":"10.1016\/j.patcog.2008.12.008","volume":"42","author":"K Noponen","year":"2009","unstructured":"Noponen K, Kortelainen J, Sepp\u00e4nen T (2009) Invariant trajectory classification of dynamical systems with a case study on ECG. Pattern Recognit 42:1832\u20131844","journal-title":"Pattern Recognit"},{"key":"6142_CR5","doi-asserted-by":"publisher","first-page":"401","DOI":"10.3109\/03091902.2013.819946","volume":"37","author":"AS Al-Fahoum","year":"2013","unstructured":"Al-Fahoum AS, Qasaimeh AM (2013) A practical reconstructed phase space approach for ECG arrhythmias classification. J Med Eng Technol 37:401\u2013408","journal-title":"J Med Eng Technol"},{"key":"6142_CR6","doi-asserted-by":"publisher","first-page":"7888","DOI":"10.3390\/e17127848","volume":"17","author":"W Le","year":"2015","unstructured":"Le W, Jiazhong Z, Wenfan Z (2015) Identify the rotating stall in centrifugal compressors by fractal dimension in reconstructed phase space. Entropy 17:7888\u20137899","journal-title":"Entropy"},{"key":"6142_CR7","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1007\/s11063-018-9965-7","volume":"50","author":"W Zeng","year":"2019","unstructured":"Zeng W, Ismail SA, Lim YP et al (2019) Classification of gait patterns using kinematic and kinetic features, gait dynamics and neural networks in patients with unilateral anterior cruciate ligament deficiency. Neural Process Lett 50:887\u2013909","journal-title":"Neural Process Lett"},{"key":"6142_CR8","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s12293-010-0042-7","volume":"2","author":"A Thammano","year":"2010","unstructured":"Thammano A, Ruxpakawong P (2010) Nonlinear dynamic system identification using recurrent neural network with multi-segment piecewise-linear connection weight. Memet Comput 2:273\u2013282","journal-title":"Memet Comput"},{"key":"6142_CR9","doi-asserted-by":"publisher","first-page":"4171","DOI":"10.1109\/TIE.2013.2288196","volume":"61","author":"H Zhao","year":"2014","unstructured":"Zhao H, Gao S, He Z et al (2014) Identification of nonlinear dynamic system using a novel recurrent wavelet neural network based on the pipelined architecture. IEEE Trans Ind Electron 61:4171\u20134182","journal-title":"IEEE Trans Ind Electron"},{"key":"6142_CR10","doi-asserted-by":"publisher","first-page":"3380","DOI":"10.1109\/TCYB.2017.2710284","volume":"47","author":"T Zheng","year":"2017","unstructured":"Zheng T, Wang C (2017) Relationship between persistent excitation levels and RBF network structures, with application to performance analysis of deterministic learning. IEEE Trans Cybern 47:3380\u20133392","journal-title":"IEEE Trans Cybern"},{"key":"6142_CR11","doi-asserted-by":"publisher","first-page":"2399","DOI":"10.1109\/TNNLS.2015.2473686","volume":"27","author":"H Ning","year":"2016","unstructured":"Ning H, Qing G, Jing X (2016) Identification of nonlinear spatiotemporal dynamical systems with nonuniform observations using reproducing-kernel-based integral least square regulation. IEEE Trans Neural Netw Learn Syst 27:2399\u20132412","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"6142_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/72.896792","volume":"12","author":"RJ Schilling","year":"2001","unstructured":"Schilling RJ, Carroll JJ, Al-Ajlouni AF (2001) Approximation of nonlinear systems with radial basis function neural networks. IEEE Trans Neural Netw 12:1\u201315","journal-title":"IEEE Trans Neural Netw"},{"key":"6142_CR13","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1109\/TNN.2007.909842","volume":"19","author":"J Lian","year":"2008","unstructured":"Lian J, Lee Y, Sudhoff SD et al (2008) Self-organizing radial basis function network for real-time approximation of continuous-time dynamical systems. IEEE Trans Neural Netw 19:460\u201374","journal-title":"IEEE Trans Neural Netw"},{"key":"6142_CR14","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1109\/TNNLS.2015.2513902","volume":"28","author":"Z Lu","year":"2017","unstructured":"Lu Z, Sun J, Butts K (2017) Multiscale support vector learning with projection operator wavelet kernel for nonlinear dynamical system identification. IEEE Trans Neural Netw Learn Syst 28:231\u2013243","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"6142_CR15","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s11063-012-9218-0","volume":"35","author":"JJ Cordova","year":"2012","unstructured":"Cordova JJ, Yu W (2012) Two types of Haar wavelet neural networks for nonlinear system identification. Neural Process Lett 35:283\u2013300","journal-title":"Neural Process Lett"},{"key":"6142_CR16","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1109\/TSMCB.2012.2210038","volume":"43","author":"Y Zhang","year":"2013","unstructured":"Zhang Y, Mu B, Zheng H (2013) Link between and comparison and combination of Zhang neural network and quasi-Newton BFGS method for time-varying quadratic minimization. IEEE Trans Cybern 43:490\u2013503","journal-title":"IEEE Trans Cybern"},{"key":"6142_CR17","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1007\/s11075-012-9690-7","volume":"64","author":"Y Zhang","year":"2013","unstructured":"Zhang Y, Li Z, Guo D et al (2013) Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer Algorithms 64:721\u2013740","journal-title":"Numer Algorithms"},{"key":"6142_CR18","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1109\/TNNLS.2014.2342260","volume":"26","author":"L Jin","year":"2017","unstructured":"Jin L, Zhang Y (2017) Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation. IEEE Trans Neural Netw Learn Syst 26:1525\u20131531","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"6142_CR19","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1007\/s00521-008-0229-8","volume":"18","author":"ED \u00dcbeyli","year":"2009","unstructured":"\u00dcbeyli ED (2009) Detecting variabilities of ECG signals by Lyapunov exponents. Neural Comput Appl 18:653\u2013662","journal-title":"Neural Comput Appl"},{"key":"6142_CR20","doi-asserted-by":"publisher","first-page":"1192","DOI":"10.1016\/j.eswa.2009.06.022","volume":"37","author":"ED Uebeyli","year":"2010","unstructured":"Uebeyli ED (2010) Recurrent neural networks employing Lyapunov exponents for analysis of ECG signals. Expert Syst Appl 37:1192\u20131199","journal-title":"Expert Syst Appl"},{"key":"6142_CR21","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1114\/B:ABME.0000042223.87320.de","volume":"32","author":"JI Salisbury","year":"2004","unstructured":"Salisbury JI, Sun Y (2004) Assessment of chaotic parameters in nonstationary electrocardiograms by use of empirical mode decomposition. Ann Biomed Eng 32:1348\u20131354","journal-title":"Ann Biomed Eng"},{"key":"6142_CR22","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s10827-020-00740-x","volume":"48","author":"RJ Cubero","year":"2020","unstructured":"Cubero RJ, Marsili M, Roudi Y et al (2020) Multiscale relevance and informative encoding in neuronal spike trains. J Comput Neurosci 48:85\u2013102","journal-title":"J Comput Neurosci"},{"key":"6142_CR23","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1038\/nrn1668","volume":"6","author":"RB Stein","year":"2005","unstructured":"Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6:389\u2013397","journal-title":"Nat Rev Neurosci"},{"key":"6142_CR24","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1109\/72.557660","volume":"8","author":"Q Zhang","year":"1997","unstructured":"Zhang Q (1997) Using wavelet network in nonparametric estimation. IEEE Trans Neural Netw 8:227\u201336","journal-title":"IEEE Trans Neural Netw"},{"key":"6142_CR25","doi-asserted-by":"publisher","first-page":"1423","DOI":"10.1109\/TPWRS.2008.922251","volume":"23","author":"NM Pindoriya","year":"2008","unstructured":"Pindoriya NM, Singh SN, Singh SK (2008) An adaptive wavelet neural network-based energy price forecasting in electricity markets. IEEE Trans Power Syst 23:1423\u20131432","journal-title":"IEEE Trans Power Syst"},{"key":"6142_CR26","doi-asserted-by":"publisher","first-page":"862","DOI":"10.1109\/TNN.2005.849842","volume":"16","author":"SA Billings","year":"2005","unstructured":"Billings SA, Wei HL (2005) A new class of wavelet networks for nonlinear system identification. IEEE Trans Neural Netw 16:862\u2013874","journal-title":"IEEE Trans Neural Netw"},{"key":"6142_CR27","doi-asserted-by":"publisher","first-page":"1359","DOI":"10.1109\/TFUZZ.2013.2286837","volume":"22","author":"YJ Liu","year":"2014","unstructured":"Liu YJ, Tong S (2014) Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash. IEEE Trans Fuzzy Syst 22:1359\u20131365","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"6142_CR28","first-page":"319","volume":"40","author":"D Kreiseler","year":"1995","unstructured":"Kreiseler D, Bousseliot R (1995) Automatisierte EKG-Auswertung mit Hilfe der EKG-Signaldatenbank CARDIODAT der PTB. Biomedizinische Technik\/Biomed Eng 40:319\u2013320","journal-title":"Biomedizinische Technik\/Biomed Eng"},{"key":"6142_CR29","first-page":"215","volume":"101","author":"AL Goldberger","year":"2020","unstructured":"Goldberger AL, Amaral LA, Glass L, Hausdorff JM, Ivanov PC, Mark RG, Mietus JE, Moody GB, Peng CK, Stanley HE (2020) Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation 101:215\u2013220","journal-title":"Circulation"},{"key":"6142_CR30","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1109\/51.870238","volume":"19","author":"Z Wang","year":"2000","unstructured":"Wang Z, Ning X, Zhang Y et al (2000) Nonlinear dynamic characteristics analysis of synchronous 12-lead ECG signals. IEEE Eng Med Biol Mag Qu Mag Eng Med Biol Soc 19:110\u2013115","journal-title":"IEEE Eng Med Biol Mag Qu Mag Eng Med Biol Soc"},{"key":"6142_CR31","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1016\/j.amjcard.2016.11.028","volume":"119","author":"M Deng","year":"2017","unstructured":"Deng M, Tang M, Wang C et al (2017) Cardiodynamicsgram as a new diagnostic tool in coronary artery disease patients with nondiagnostic electrocardiograms. Am J Cardiol 119:698\u2013704","journal-title":"Am J Cardiol"},{"key":"6142_CR32","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1109\/82.246163","volume":"40","author":"KM Cuomo","year":"2002","unstructured":"Cuomo KM, Oppenheim AV, Strogatz SH (2002) Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circuits Syst II Analog Digital Signal Process 40:626\u2013633","journal-title":"IEEE Trans Circuits Syst II Analog Digital Signal Process"},{"key":"6142_CR33","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1080\/00207720410001703213","volume":"35","author":"A Al-Ajlouni","year":"2004","unstructured":"Al-Ajlouni A, Schilling R, Harris S (2004) Synchronization of Lorenz-based chaotic circuits with applications to communications. Int J Syst Sci 35:211\u2013221","journal-title":"Int J Syst Sci"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-021-06142-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-021-06142-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-021-06142-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,24]],"date-time":"2021-10-24T11:23:46Z","timestamp":1635074626000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-021-06142-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,16]]},"references-count":33,"journal-issue":{"issue":"22","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["6142"],"URL":"https:\/\/doi.org\/10.1007\/s00521-021-06142-z","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,16]]},"assertion":[{"value":"11 August 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declaration"}},{"value":"The authors (Guo Luo, Zhi Yang and Qizhi Zhang) declare that they have no conflict of interests in relation to the work in this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}