{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:24:41Z","timestamp":1760235881435,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,8]],"date-time":"2021-10-08T00:00:00Z","timestamp":1633651200000},"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>Previous studies have demonstrated, experimentally and theoretically, the existence of slow\u2013fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow\u2013fast regime.<\/jats:p>","DOI":"10.3390\/sym13101898","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1898","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Slow Invariant Manifold of Laser with Feedback"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1400-4136","authenticated-orcid":false,"given":"Jean-Marc","family":"Ginoux","sequence":"first","affiliation":[{"name":"Center for Theoretical Physics, CNRS, UMR 7332, Universit\u00e9 de Toulon, CS 60584, CEDEX 9, 83041 Toulon, France"}]},{"given":"Riccardo","family":"Meucci","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, 50125 Firenze, Italy"},{"name":"Department of Physics and Astronomy, Universit\u00e0 di Firenze, 50125 Firenze, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"073022","DOI":"10.1088\/1367-2630\/11\/7\/073022","article-title":"Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback","volume":"11","author":"Marino","year":"2009","journal-title":"New J. Phys."},{"key":"ref_2","unstructured":"Poincar\u00e9, H. (1892). Les M\u00e9thodes Nouvelles de la M\u00e9canique C\u00e9leste, Gauthier-Villars."},{"key":"ref_3","unstructured":"Andronov, A.A., and Chaikin, S.E. (1949). Theory of Oscillators, Moscow, I., English Translation, Princeton University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"127","DOI":"10.2307\/1969357","article-title":"A second-order differential equation with singular solutions","volume":"50","author":"Levinson","year":"1949","journal-title":"Ann. Math."},{"key":"ref_5","first-page":"575","article-title":"On the dependence of solutions of differential equations on a small parameter","volume":"31","author":"Tikhonov","year":"1948","journal-title":"Mat. Sb. N.S."},{"key":"ref_6","unstructured":"Wasow, W.R. (1965). Asymptotic Expansions for Ordinary Differential Equations, Wiley-Interscience."},{"key":"ref_7","unstructured":"Cole, J.D. (1968). Perturbation Methods in Applied Mathematics, Blaisdell."},{"key":"ref_8","unstructured":"O\u2019Malley, R.E. (1974). Introduction to Singular Perturbations, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"O\u2019Malley, R.E. (1991). Singular Perturbations Methods for Ordinary Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-0977-5"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1512\/iumj.1972.21.21017","article-title":"Persistence and Smoothness of Invariant Manifolds for Flows","volume":"21","author":"Fenichel","year":"1971","journal-title":"Ind. Univ. Math. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1512\/iumj.1974.23.23090","article-title":"Asymptotic stability with rate conditions","volume":"23","author":"Fenichel","year":"1974","journal-title":"Ind. Univ. Math. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1512\/iumj.1977.26.26006","article-title":"Asymptotic stability with rate conditions II","volume":"26","author":"Fenichel","year":"1977","journal-title":"Ind. Univ. Math. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/0022-0396(79)90152-9","article-title":"Geometric singular perturbation theory for ordinary differential equations","volume":"31","author":"Fenichel","year":"1979","journal-title":"J. Diff. Eq."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hirsch, M.W., Pugh, C.C., and Shub, M. (1977). Invariant Manifolds, Springer.","DOI":"10.1007\/BFb0092042"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rossetto, B. (1986). Trajectoires lentes des syst\u2018emes dynamiques lents-rapides. Analysis and Optimization of System, Springer.","DOI":"10.1007\/BFb0007600"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rossetto, B. (1987). Singular approximation of chaotic slow-fast dynamical systems. The Physics of Phase Space Nonlinear Dynamics and Chaos Geometric Quantization, and Wigner Function, Springer.","DOI":"10.1007\/3-540-17894-5_306"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1137\/040608295","article-title":"Projecting to a slow manifold: Singularly perturbed systems and legacy codes","volume":"4","author":"Gear","year":"2005","journal-title":"SIAM J. Appl. Dyn. Syst. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1051\/m2an\/2009026","article-title":"Analysis of the accuracy and convergence of equation-free projection to a slow manifold","volume":"43","author":"Zagaris","year":"2009","journal-title":"ESAIM Math. Model. Num."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0010-2180(92)90034-M","article-title":"Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space","volume":"8","author":"Maas","year":"1992","journal-title":"Combust. Flame"},{"key":"ref_20","first-page":"305","article-title":"Asymptotic analysis of canards in the EOE equations and the role of the inflection line","volume":"445","year":"1994","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1142\/S0218127498001765","article-title":"Slow fast autonomous dynamical systems","volume":"8","author":"Rossetto","year":"1998","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Aziz-Alaoui, M.A., and Bertelle, C. (2006). Slow manifold of a neuronal bursting model. Emergent Properties in Natural and Articial Dynamical Systems, Springer.","DOI":"10.1007\/3-540-34824-7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1142\/S0218127406015192","article-title":"Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems","volume":"4","author":"Ginoux","year":"2006","journal-title":"Int. J. Bif. Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3409","DOI":"10.1142\/S0218127408022457","article-title":"Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems","volume":"11","author":"Ginoux","year":"2008","journal-title":"Int. J. Bif. Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ginoux, J.M. (2009). Differential Geometry Applied to Dynamical Systems, World Scientific.","DOI":"10.1142\/9789814277150"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"465203","DOI":"10.1088\/1751-8113\/44\/46\/465203","article-title":"The flow curvature method applied to canard explosion","volume":"44","author":"Ginoux","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s12346-013-0104-6","article-title":"The Slow Invariant Manifold of the Lorenz-Krishnamurthy Model","volume":"13","author":"Ginoux","year":"2014","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2150112-1-17","DOI":"10.1142\/S0218127421501121","article-title":"Slow Invariant Manifolds of Slow-Fast Dynamical Systems","volume":"31","author":"Ginoux","year":"2021","journal-title":"Int. J. Bif. Chaos"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bender, C.M., and Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers, Springer.","DOI":"10.1007\/978-1-4757-3069-2"},{"key":"ref_30","first-page":"78","article-title":"Numerical Calculation of Lyapunov Exponents","volume":"6","author":"Sandri","year":"1996","journal-title":"Math. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov Exponents from a Time Series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1103\/RevModPhys.57.617","article-title":"Ergodic Theory of Chaos and Strange Attractors","volume":"57","author":"Eckmann","year":"1985","journal-title":"Rev. Mod. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Baier, G., and Klein, M. (1991). Hierarchies of Dynamical Systems. A Chaotic Hierarchy, World Scientific.","DOI":"10.1142\/0934"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1330010","DOI":"10.1142\/S0218127413300103","article-title":"Canards from Chua\u2019s circuit","volume":"23","author":"Ginoux","year":"2013","journal-title":"Int. J. Bif. Chaos"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"342010","DOI":"10.1155\/2015\/342010","article-title":"Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models","volume":"2015","author":"Ginoux","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s12346-015-0160-1","article-title":"Canards Existence in Memristor\u2019s Circuits","volume":"15","author":"Ginoux","year":"2016","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1051\/mmnp\/2019012","article-title":"Canards Existence in The Hindmarsh-Rose Model","volume":"14","author":"Ginoux","year":"2019","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_38","first-page":"60","article-title":"M\u00e9moire sur les \u00e9quations diff\u00e9rentielles alg\u00e9briques du premier ordre et du premier degr\u00e9","volume":"2","author":"Darboux","year":"1878","journal-title":"Bull. Sci. Math. 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