{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T02:35:26Z","timestamp":1780454126918,"version":"3.54.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2018,5,11]],"date-time":"2018-05-11T00:00:00Z","timestamp":1525996800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100007136","name":"Secretar\u00eda de Estado de Investigaci\u00f3n, Desarrollo e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["TRA2016-75075-R"],"award-info":[{"award-number":["TRA2016-75075-R"]}],"id":[{"id":"10.13039\/501100007136","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2018,10]]},"DOI":"10.1007\/s00332-018-9464-z","type":"journal-article","created":{"date-parts":[[2018,5,11]],"date-time":"2018-05-11T07:15:25Z","timestamp":1526022925000},"page":"1793-1842","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Spatio-Temporal Koopman Decomposition"],"prefix":"10.1007","volume":"28","author":[{"given":"Soledad Le","family":"Clainche","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 M.","family":"Vega","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2018,5,11]]},"reference":[{"key":"9464_CR1","unstructured":"Andersson, N., Eriksson, L.-E.: A novel solver acceleration technique based on dynamic mode decomposition. In: 6th European Conference on Computational Fluid Dynamics (ECFD VI), Barcelona, Spain, pp. 4832\u20134851, 20\u201325 July (2014)"},{"key":"9464_CR2","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1103\/RevModPhys.74.99","volume":"74","author":"IS Aranson","year":"2002","unstructured":"Aranson, I.S., Kramer, L.: The world of the complex Ginzburg\u2013Landau equation. Rev. Mod. Phys. 74, 100\u2013142 (2002)","journal-title":"Rev. Mod. Phys."},{"key":"9464_CR3","unstructured":"Arbabi, H., Mezi\u0107, I.: Ergodic theory, dynamic mode decomposition and computation of spectral properties of the Koopman operator (2016). arXiv preprint arXiv:1611.06664"},{"key":"9464_CR4","doi-asserted-by":"publisher","first-page":"1315","DOI":"10.1016\/j.conengprac.2011.07.004","volume":"19","author":"N Benito","year":"2011","unstructured":"Benito, N., Arias, J.R., Velazquez, A., Vega, J.M.: Real time performance improvement of engineering control units via higher order singular value decomposition: application to a SI engine. Control Eng. Pract. 19, 1315\u20131327 (2011)","journal-title":"Control Eng. Pract."},{"key":"9464_CR5","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1017\/S0022112004002095","volume":"522","author":"HM Blackburn","year":"2005","unstructured":"Blackburn, H.M., Marques, F., Lopez, J.M.: Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395\u2013411 (2005)","journal-title":"J. Fluid Mech."},{"key":"9464_CR6","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1038\/s41467-017-00030-8","volume":"8","author":"SL Brunton","year":"2017","unstructured":"Brunton, S.L., Brunton, B.W., Proctor, J.L., Kaiser, E., Kutz, J.N.: Chaos as an intermittently forced linear system. Nat. Commun. 8, 19 (2017)","journal-title":"Nat. Commun."},{"key":"9464_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-06406-1","volume-title":"Convective Heat Transfer","author":"T Cebeci","year":"2002","unstructured":"Cebeci, T.: Convective Heat Transfer. Springer, Berlin (2002)"},{"key":"9464_CR8","volume-title":"Hydrodynamic and Hydromagnetic Stability","author":"S Chandrasekhar","year":"1961","unstructured":"Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)"},{"key":"9464_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-32093-4","volume-title":"Solar and Stellar Dynamos","author":"P Charbonneau","year":"2013","unstructured":"Charbonneau, P.: Solar and Stellar Dynamos. Springer, Berlin (2013)"},{"key":"9464_CR10","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1103\/RevModPhys.65.851","volume":"65","author":"M Cross","year":"1993","unstructured":"Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851\u20131112 (1993)","journal-title":"Rev. Mod. Phys."},{"key":"9464_CR11","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1137\/S0895479896305696","volume":"21","author":"L Lathauwer de","year":"2000","unstructured":"de Lathauwer, L., de Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253\u20131278 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9464_CR12","doi-asserted-by":"publisher","first-page":"2713","DOI":"10.2514\/1.J051133","volume":"49","author":"S Lucas de","year":"2011","unstructured":"de Lucas, S., Vega, J.M., Velazquez, A.: Aeronautic conceptual design optimization method based on high order singular value decomposition. AIAA J. 49, 2713\u20132725 (2011)","journal-title":"AIAA J."},{"key":"9464_CR13","doi-asserted-by":"publisher","first-page":"7997","DOI":"10.1016\/j.jcp.2010.07.004","volume":"229","author":"F Garcia","year":"2010","unstructured":"Garcia, F., Net, M., Garcia-Archilla, B., S\u00e1nchez, J.: A comparison of high-order time integrators for thermal convection in rotating spherical shells. J. Comput. Phys. 229, 7997\u20138010 (2010)","journal-title":"J. Comput. Phys."},{"key":"9464_CR14","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.pepi.2014.02.004","volume":"230","author":"F Garcia","year":"2014","unstructured":"Garcia, F., S\u00e1nchez, J., Net, M.: Numerical simulations of thermal convection in rotating spherical shells under laboratory conditions. Phys. Earth Planet. Inter. 230, 28\u201344 (2014)","journal-title":"Phys. Earth Planet. Inter."},{"key":"9464_CR15","first-page":"103","volume":"44","author":"D Giannakis","year":"2015","unstructured":"Giannakis, D., Slawinska, J., Zhao, Z.: Spatio-temporal feature extraction with data-driven Koopman operators. J. Mach. Learn. Res. Workshop Conf. Proc. 44, 103\u2013115 (2015)","journal-title":"J. Mach. Learn. Res. Workshop Conf. Proc."},{"key":"9464_CR16","volume-title":"Matrix Computations","author":"GH Golub","year":"1996","unstructured":"Golub, G.H., van Loan, G.T.: Matrix Computations. John Hopkins University Press, Baltimore (1996)"},{"key":"9464_CR17","volume-title":"Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems","author":"M Haragus","year":"2010","unstructured":"Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems. Springer, Berlin (2010)"},{"key":"9464_CR18","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455\u2013500 (2009)","journal-title":"SIAM Rev."},{"key":"9464_CR19","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1073\/pnas.17.5.315","volume":"17","author":"B Koopman","year":"1931","unstructured":"Koopman, B.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. USA 17, 315\u2013318 (1931)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"9464_CR20","doi-asserted-by":"crossref","unstructured":"Le Clainche, S., Sastre, F., Vega, J.M., Velazquez, A.: Higher order dynamic mode decomposition of noisy experimental data: the flow structure of a zero-net-mass-flux jet. In: Proceedings of 47th AIAA Fluid Dynamics Conference, AIAA paper 2017\u20133304 (2017)","DOI":"10.1016\/j.expthermflusci.2017.06.011"},{"issue":"2","key":"9464_CR21","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1137\/15M1054924","volume":"16","author":"S Clainche Le","year":"2017","unstructured":"Le Clainche, S., Vega, J.M.: Higher order dynamic mode decomposition. SIAM J. Appl. Dyn. Syst. 16(2), 882\u2013925 (2017)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"8","key":"9464_CR22","doi-asserted-by":"publisher","first-page":"084102","DOI":"10.1063\/1.4997206","volume":"29","author":"S Clainche Le","year":"2017","unstructured":"Le Clainche, S., Vega, J.M.: Higher order dynamic mode decomposition to identify and extrapolate flow patterns. Phys. Fluids 29(8), 084102 (2017)","journal-title":"Phys. Fluids"},{"key":"9464_CR23","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.expthermflusci.2017.06.011","volume":"88","author":"S Clainche Le","year":"2017","unstructured":"Le Clainche, S., Vega, J.M., Soria, J.: Higher order dynamic mode decomposition of noisy experimental data: the flow structure of a zero-net-mass-flux jet. Exp. Therm. Fluid Sci. 88, 336\u2013353 (2017)","journal-title":"Exp. Therm. Fluid Sci."},{"key":"9464_CR24","doi-asserted-by":"publisher","first-page":"1779","DOI":"10.2514\/1.35258","volume":"45","author":"LS Lorente","year":"2008","unstructured":"Lorente, L.S., Vega, J.M., Velazquez, A.: Generation of aerodynamics databases using high-order singular value decomposition. J. Aircraft 45, 1779\u20131788 (2008)","journal-title":"J. Aircraft"},{"key":"9464_CR25","doi-asserted-by":"publisher","first-page":"20120348","DOI":"10.1098\/rspa.2012.0348","volume":"469","author":"F Marques","year":"2013","unstructured":"Marques, F., Meseguer, A., Lopez, J.M., Pacheco, J.R., Lopez, J.M.: Bifurcations with imperfect SO(2) symmetry and pinning of rotating waves. Proc. R. Soc. A 469, 20120348 (2013)","journal-title":"Proc. R. Soc. A"},{"key":"9464_CR26","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1017\/S0022112010005239","volume":"670","author":"F Mellibovsky","year":"2011","unstructured":"Mellibovsky, F., Eckhardt, B.: Takens\u2013Bogdanov bifurcation of travelling-wave solutions in pipe flow. J. Fluid Mech. 670, 96\u2013129 (2011)","journal-title":"J. Fluid Mech."},{"issue":"1","key":"9464_CR27","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s11071-005-2824-x","volume":"41","author":"I Mezi\u0107","year":"2005","unstructured":"Mezi\u0107, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309\u2013325 (2005)","journal-title":"Nonlinear Dyn."},{"key":"9464_CR28","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1146\/annurev-fluid-011212-140652","volume":"45","author":"I Mezi\u0107","year":"2013","unstructured":"Mezi\u0107, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Ann. Rev. Fluid Mech. 45, 357\u2013378 (2013)","journal-title":"Ann. Rev. Fluid Mech."},{"key":"9464_CR29","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.physd.2004.06.015","volume":"197","author":"I Mezi\u0107","year":"2004","unstructured":"Mezi\u0107, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197, 101\u2013133 (2004)","journal-title":"Physica D"},{"key":"9464_CR30","doi-asserted-by":"publisher","first-page":"18371","DOI":"10.1073\/pnas.0709146104","volume":"104","author":"L Omberg","year":"2007","unstructured":"Omberg, L., Golub, G.H., Alter, O.: A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies. Proc. Nat. Acad. USA 104, 18371\u201318376 (2007)","journal-title":"Proc. Nat. Acad. USA"},{"key":"9464_CR31","volume-title":"Numerical Recipes in C","author":"WH Press","year":"1988","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge Univeristy Press, Cambridge (1988)"},{"key":"9464_CR32","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1017\/S0022112009992059","volume":"641","author":"CW Rowley","year":"2009","unstructured":"Rowley, C.W., Mezi\u0107, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115\u2013127 (2009)","journal-title":"J. Fluid Mech."},{"key":"9464_CR33","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/j.physd.2009.10.012","volume":"239","author":"J Sanchez","year":"2010","unstructured":"Sanchez, J., Net, M., Simo, C.: Computation of invariant tori by Newton\u2013Krylov methods in large-scale dissipative systems. Physica D 239, 123\u2013133 (2010)","journal-title":"Physica D"},{"issue":"033014","key":"9464_CR34","first-page":"1","volume":"87","author":"J S\u00e1nchez","year":"2013","unstructured":"S\u00e1nchez, J., Garc\u00eda, F., Net, M.: Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation. Phys. Rev. E 87(033014), 1\u201311 (2013)","journal-title":"Phys. Rev. E"},{"key":"9464_CR35","doi-asserted-by":"publisher","first-page":"993","DOI":"10.1016\/j.patcog.2006.08.004","volume":"40","author":"B Savas","year":"2007","unstructured":"Savas, B., Eden, L.: Handwritten digit classification using higher order singular value decomposition. Pattern Recognit. 40, 993\u20131003 (2007)","journal-title":"Pattern Recognit."},{"key":"9464_CR36","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1017\/S0022112010001217","volume":"656","author":"PJ Schmid","year":"2010","unstructured":"Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5\u201328 (2010)","journal-title":"J. Fluid Mech."},{"key":"9464_CR37","doi-asserted-by":"publisher","first-page":"032402","DOI":"10.1103\/PhysRevFluids.1.032402","volume":"1","author":"AS Sharma","year":"2016","unstructured":"Sharma, A.S., Mezi\u0107, I., McKeon, B.J.: Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier\u2013Stokes equations. Phys. Rev. Fluids 1, 032402 (2016)","journal-title":"Phys. Rev. Fluids"},{"key":"9464_CR38","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1137\/1015095","volume":"15","author":"GW Stewart","year":"1973","unstructured":"Stewart, G.W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15, 727\u2013764 (1973)","journal-title":"SIAM Rev."},{"key":"9464_CR39","doi-asserted-by":"crossref","unstructured":"Susuki, Y., Mezi\u0107, I.: A prony approximation of Koopman mode decomposition. In: 2015 IEEE 54th Annual Conference Decision and Control (CDC), pp. 7022\u20137027. IEEE (2015)","DOI":"10.1109\/CDC.2015.7403326"},{"key":"9464_CR40","first-page":"366","volume-title":"Lecture Notes in Mathematics","author":"F Takens","year":"1981","unstructured":"Takens, F.: Detecting strange attractors in turbulence. In: Rand, D.A., Young, L.-S. (eds.) Lecture Notes in Mathematics, pp. 366\u2013381. Springer, Berlin (1981)"},{"key":"9464_CR41","unstructured":"Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications (2013). arXiv preprint: arXiv:1312.0041"},{"key":"9464_CR42","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/BF02289464","volume":"3","author":"LR Tucker","year":"1966","unstructured":"Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 3, 279\u2013311 (1966)","journal-title":"Psychometrika"},{"key":"9464_CR43","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/0-306-47815-3_5","volume-title":"A Practical Approach to Microarray Data Analysis","author":"ME Wall","year":"2003","unstructured":"Wall, M.E., Rechtsteiner, A., Rocha, L.M.: Singular value decomposition and principal component analysis. In: Berrar, D.P., Dubitzky, W., Granzowr, M. (eds.) A Practical Approach to Microarray Data Analysis, pp. 91\u2013109. Kluwer, Norwell (2003)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-018-9464-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-018-9464-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-018-9464-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,23]],"date-time":"2022-08-23T11:52:33Z","timestamp":1661255553000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-018-9464-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,11]]},"references-count":43,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,10]]}},"alternative-id":["9464"],"URL":"https:\/\/doi.org\/10.1007\/s00332-018-9464-z","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,11]]},"assertion":[{"value":"26 April 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}