{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:37:09Z","timestamp":1760240229179,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,10]],"date-time":"2019-04-10T00:00:00Z","timestamp":1554854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were obtained. Furthermore, computer calculations for several parameter configurations were made; rich dynamic behavior among periodic orbits high periodicity and chaos were observed through bifurcation diagrams. These results confirm the direct dependence between the parameters present in the intracavity chaos-generating element.<\/jats:p>","DOI":"10.3390\/e21040384","type":"journal-article","created":{"date-parts":[[2019,4,10]],"date-time":"2019-04-10T11:25:08Z","timestamp":1554895508000},"page":"384","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0276-6774","authenticated-orcid":false,"given":"Vicente","family":"Aboites","sequence":"first","affiliation":[{"name":"Centro de Investigaciones en \u00d3ptica, Loma del Bosque 115, 37150 Le\u00f3n, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Liceaga","sequence":"additional","affiliation":[{"name":"Divisi\u00f3n de Ciencias e Ingenier\u00edas, Universidad de Guanajuato, Loma del Bosque 107, 37150 Le\u00f3n, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8137-1270","authenticated-orcid":false,"given":"Rider","family":"Jaimes-Re\u00e1tegui","sequence":"additional","affiliation":[{"name":"Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Diaz de Le\u00f3n 1144, Paseos de la Monta\u00f1a, Lagos de Moreno, 47460 Jalisco, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3739-0781","authenticated-orcid":false,"given":"Juan Hugo","family":"Garc\u00eda-L\u00f3pez","sequence":"additional","affiliation":[{"name":"Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Diaz de Le\u00f3n 1144, Paseos de la Monta\u00f1a, Lagos de Moreno, 47460 Jalisco, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1364\/OL.16.000196","article-title":"Beam propagation and the ABCD ray matrices","volume":"16","year":"1991","journal-title":"Opt. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1088\/0150-536X\/23\/4\/004","article-title":"ABCD propagation law for misaligned general astigmatic Gaussian beams","volume":"23","author":"Onciul","year":"1992","journal-title":"J. Opt."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"S1011","DOI":"10.1007\/BF01588602","article-title":"ABCD law for partially coherent Gaussian light, propagating through first-order optical systems","volume":"24","author":"Bastiaans","year":"1992","journal-title":"Opt. Quantum Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4148","DOI":"10.1364\/OL.35.004148","article-title":"Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function","volume":"35","author":"Tian","year":"2010","journal-title":"Opt. Lett."},{"key":"ref_5","unstructured":"Siegman, A.E. (1986). Lasers, University Science Books."},{"key":"ref_6","unstructured":"Tarasov, L. (1981). Physique Des Processus Dans Les Generaeurs De Rayonnement Optique Coherent, Mir, Moscu. (trans. from Russian)."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/amis\/100608","article-title":"Ikeda Map and Phase Conjugated Ring Resonator Chaotic Dynamics","volume":"10","author":"Aboites","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_8","first-page":"2823","article-title":"Standard Map Spatial Dynamics in a Ring-Phase Conjugated Resonator","volume":"9","author":"Aboites","year":"2015","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.rinp.2012.11.001","article-title":"Tinkerbell beams in a non-linear ring resonator","volume":"2","author":"Aboites","year":"2012","journal-title":"Results Phys."},{"key":"ref_10","first-page":"429","article-title":"Tinkerbell chaos in a ring phase-conjugated resonator","volume":"54","author":"Aboites","year":"2009","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_11","first-page":"13","article-title":"Bidimensional dynamics maps in optical resonators","volume":"60","author":"Aboites","year":"2014","journal-title":"Rev. Mex. Fis."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"076002","DOI":"10.1088\/1054-660X\/23\/7\/075002","article-title":"Duffing spatial dynamics induced in a double phase-conjugated resonator","volume":"23","author":"Dignowity","year":"2013","journal-title":"Laser Phys."},{"key":"ref_13","first-page":"345","article-title":"Dynamics of a LASER Resonator","volume":"36","author":"Aboites","year":"2007","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_14","first-page":"129","article-title":"Henon beams","volume":"65","author":"Aboites","year":"2010","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1142\/S021812749300074X","article-title":"The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System","volume":"3","author":"Arrowsmith","year":"1993","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1119\/1.1970159","article-title":"Matrix Representation of Gaussian Optics","volume":"32","author":"Hallbach","year":"1964","journal-title":"Am. J. Phys."},{"key":"ref_17","first-page":"373","article-title":"Versal deformations of a singularity of a vector field on the plane in the case of zero eigenvalues","volume":"1","author":"Bogdanov","year":"1981","journal-title":"Sel. Math. Sov."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/384\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:44:18Z","timestamp":1760186658000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/384"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,10]]},"references-count":17,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["e21040384"],"URL":"https:\/\/doi.org\/10.3390\/e21040384","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,4,10]]}}}