{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:26:29Z","timestamp":1760228789668,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,25]],"date-time":"2022-05-25T00:00:00Z","timestamp":1653436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we reconstruct the dynamic behavior of the ring-coupled Lorenz oscillators system by reservoir computing. Although the reconstruction of various complex chaotic attractors has been well studied by using various neural networks, little attention has been paid to whether the spatio-temporal structure of some special attractors can be maintained in long-term prediction. Reservoir computing has been shown to be effective for model-free prediction, so we want to investigate whether reservoir computing can restore the rotational symmetry of the original ring-coupled Lorenz system. We find that although the state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated spatio-temporal structure is maintained in the process of reconstruction. Specifically, we show that the rotational symmetric structure of periodic rotating waves, quasi-periodic torus, and chaotic rotating waves is well maintained.<\/jats:p>","DOI":"10.3390\/sym14061084","type":"journal-article","created":{"date-parts":[[2022,5,25]],"date-time":"2022-05-25T05:12:27Z","timestamp":1653455547000},"page":"1084","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Learning Coupled Oscillators System with Reservoir Computing"],"prefix":"10.3390","volume":"14","author":[{"given":"Xijuan","family":"Zhong","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4409-0581","authenticated-orcid":false,"given":"Shuai","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130000, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1103\/PhysRevLett.64.821","article-title":"Synchronization in Chaotic Systems","volume":"64","author":"Pecora","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1103\/PhysRevLett.76.1804","article-title":"Phase Synchronization of Chaotic Oscillators","volume":"76","author":"Rosenblum","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2357","DOI":"10.1103\/PhysRevE.47.2357","article-title":"Synchronization of chaotic trajectories using control","volume":"47","author":"Lai","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(02)00137-0","article-title":"The synchronization of chaotic systems","volume":"366","author":"Boccaletti","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1143\/PTP.69.32","article-title":"Stability theory of synchronized motion in coupled-oscillator systems","volume":"69","author":"Fujisaka","year":"1983","journal-title":"Prog. Theor. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1143\/PTP.74.918","article-title":"Stability theory of synchronized motion in coupled-oscillator systems. IV","volume":"74","author":"Fujisaka","year":"1985","journal-title":"Prog. Theor. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1240","DOI":"10.1143\/PTP.70.1240","article-title":"Stability theory of synchronized motion in coupled-oscillator systems. II","volume":"70","author":"Fujisaka","year":"1983","journal-title":"Prog. Theor. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1143\/PTP.72.885","article-title":"Stability theory of synchronized motion in coupled-oscillator systems. III","volume":"72","author":"Fujisaka","year":"1984","journal-title":"Prog. Theor. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"035204","DOI":"10.1103\/PhysRevE.65.035204","article-title":"Reconstructing embedding spaces of coupled dynamical systems from multivariate data","volume":"65","author":"Boccaletti","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1126\/science.1091277","article-title":"Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication","volume":"304","author":"Jaeger","year":"2004","journal-title":"Science"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121102","DOI":"10.1063\/1.5010300","article-title":"Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data","volume":"12","author":"Pathak","year":"2017","journal-title":"Chaos"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"024102","DOI":"10.1103\/PhysRevLett.120.024102","article-title":"Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach","volume":"120","author":"Pathak","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"033314","DOI":"10.1103\/PhysRevE.102.033314","article-title":"Mapping topological characteristics of dynamical systems into neural networks: A reservoir computing approach","volume":"102","author":"Chen","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123108","DOI":"10.1063\/1.5120710","article-title":"Forecasting chaotic systems with very low connectivity reservoir computers","volume":"29","author":"Griffith","year":"2019","journal-title":"Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"043118","DOI":"10.1063\/1.5022276","article-title":"Observing spatio-temporal dynamics of excitable media using reservoir computing","volume":"28","author":"Zimmermann","year":"2018","journal-title":"Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"052209","DOI":"10.1103\/PhysRevE.98.052209","article-title":"Using reservoir computers to distinguish chaotic signals","volume":"98","author":"Carroll","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"033056","DOI":"10.1103\/PhysRevResearch.1.033056","article-title":"Model-free prediction of spatiotemporal dynamical systems with recurrent neural networks: Role of network spectral radius","volume":"1","author":"Jiang","year":"2019","journal-title":"Phys. Rev. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113119","DOI":"10.1063\/1.5119723","article-title":"Predicting slow and fast neuronal dynamics with machine learning","volume":"29","author":"Follmann","year":"2019","journal-title":"Chaos"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.cosrev.2009.03.005","article-title":"Reservoir computing approaches to recurrent neural network training","volume":"3","author":"Jaeger","year":"2009","journal-title":"Comput. Sci. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"061104","DOI":"10.1063\/1.5039508","article-title":"Attractor reconstruction by machine learning","volume":"28","author":"Lu","year":"2018","journal-title":"Chaos"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"024205","DOI":"10.1103\/PhysRevE.104.024205","article-title":"Learning Hamiltonian dynamics with reservoir computing","volume":"104","author":"Zhang","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"033122","DOI":"10.1063\/1.2335815","article-title":"Experimental study of the transitions between synchronous chaos and a periodic rotating wave","volume":"16","year":"2006","journal-title":"Chaos"},{"key":"ref_23","first-page":"2335","article-title":"Transition to rotating chaotic waves in arrays of coupled Lorenz oscillators","volume":"9","year":"1998","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105370","DOI":"10.1016\/j.cnsns.2020.105370","article-title":"The mechanism of rotating waves in a ring of unidirectionally coupled Lorenz systems","volume":"90","author":"Wang","year":"2020","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"035205","DOI":"10.1088\/1402-4896\/ac46f3","article-title":"Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types","volume":"97","author":"Wang","year":"2022","journal-title":"Phys.Scr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"133208","DOI":"10.1016\/j.physd.2022.133208","article-title":"Synchronization, symmetry and rotating periodic solutions in oscillators with Huygens\u2019 coupling","volume":"434","author":"Wang","year":"2022","journal-title":"Physica D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.physd.2013.06.003","article-title":"Metastable and chaotic transient rotating waves in a ring of unidirectionally coupled bistable Lorenz systems","volume":"261","author":"Horikawa","year":"2013","journal-title":"Physica D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7139","DOI":"10.1002\/mma.4518","article-title":"Rotating periodic solutions for asymptotically linear second-order Hamiltonian systems with resonance at infinity","volume":"40","author":"Liu","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.1016\/j.jde.2018.04.001","article-title":"Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems","volume":"265","author":"Liu","year":"2018","journal-title":"J. Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2437","DOI":"10.1216\/RMJ-2017-47-7-2423","article-title":"Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions","volume":"47","author":"Yang","year":"2017","journal-title":"Rocky Mt. J. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"097611","DOI":"10.1063\/1.4917383","article-title":"Synchronization in Chaotic Systems","volume":"25","author":"Pecora","year":"2015","journal-title":"Chaos"},{"key":"ref_32","first-page":"001","article-title":"Hybrid data-driven fuzzy active disturbance rejection control for tower crane systems","volume":"08","author":"Roman","year":"2020","journal-title":"Eur. J. Control."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Chi, R.H., Li, H.Y., Shen, D., Hou, Z.S., and Huang, B. (2022). Enhanced P-type Control: Indirect Adaptive Learning from Set-point Updates. IEEE Trans. Autom. Control.","DOI":"10.1109\/TAC.2022.3154347"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1082","DOI":"10.1103\/PhysRevLett.55.1082","article-title":"Measurement of the Lyapunov spectrum from a chaotic time series","volume":"55","author":"Sano","year":"1985","journal-title":"Phys. Rev. Lett."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1084\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:18:04Z","timestamp":1760138284000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1084"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,25]]},"references-count":34,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061084"],"URL":"https:\/\/doi.org\/10.3390\/sym14061084","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,25]]}}}