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The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase-space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications. <\/jats:p>","DOI":"10.1142\/s0218127416300111","type":"journal-article","created":{"date-parts":[[2016,5,5]],"date-time":"2016-05-05T08:18:02Z","timestamp":1462436282000},"page":"1630011","source":"Crossref","is-referenced-by-count":10,"title":["Bifurcation Sequences in the Symmetric 1:1 Hamiltonian Resonance"],"prefix":"10.1142","volume":"26","author":[{"given":"Antonella","family":"Marchesiello","sequence":"first","affiliation":[{"name":"Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, D\u011b\u010d\u00edn Branch, Pohranicn\u00ed 1, 40501 D\u011b\u010d\u00edn, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Pucacco","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica and INFN \u2013 Sezione di Roma II, Universit\u00e0 di Roma \u201cTor Vergata\u201d, Via della Ricerca Scientifica, 1 - 00133 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,5,5]]},"reference":[{"key":"S0218127416300111BIB001","volume-title":"Foundations of Mechanics","author":"Abraham R.","year":"1987","edition":"2"},{"key":"S0218127416300111BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2063-1"},{"key":"S0218127416300111BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2013.04.017"},{"key":"S0218127416300111BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s10569-006-9015-z"},{"key":"S0218127416300111BIB005","doi-asserted-by":"publisher","DOI":"10.1086\/521423"},{"key":"S0218127416300111BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03319-7"},{"key":"S0218127416300111BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09240-8"},{"key":"S0218127416300111BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(97)00202-9"},{"key":"S0218127416300111BIB009","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/11\/6\/009"},{"key":"S0218127416300111BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/s10231-015-0474-2"},{"key":"S0218127416300111BIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2015.12.004"},{"key":"S0218127416300111BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85146-2"},{"key":"S0218127416300111BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-015-9232-2"},{"key":"S0218127416300111BIB014","first-page":"49","volume":"227","author":"Contopoulos G.","year":"1990","journal-title":"Astron. 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