[{"_id":"16633","department":[{"_id":"101"}],"user_id":"15701","language":[{"iso":"eng"}],"publication":"Mathematical Proceedings of the Cambridge Philosophical Society","type":"journal_article","abstract":[{"text":"<jats:title>Abstract</jats:title><jats:p>We obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.</jats:p><jats:p>Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.</jats:p>","lang":"eng"}],"status":"public","date_updated":"2022-01-06T06:52:53Z","author":[{"last_name":"Melbourne","full_name":"Melbourne, Ian","first_name":"Ian"},{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"}],"date_created":"2020-04-16T08:31:40Z","title":"Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group","doi":"10.1017/s0305004100071577","publication_identifier":{"issn":["0305-0041","1469-8064"]},"publication_status":"published","year":"1993","page":"235-268","citation":{"bibtex":"@article{Melbourne_Dellnitz_1993, title={Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group}, DOI={<a href=\"https://doi.org/10.1017/s0305004100071577\">10.1017/s0305004100071577</a>}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, author={Melbourne, Ian and Dellnitz, Michael}, year={1993}, pages={235–268} }","short":"I. Melbourne, M. Dellnitz, Mathematical Proceedings of the Cambridge Philosophical Society (1993) 235–268.","mla":"Melbourne, Ian, and Michael Dellnitz. “Normal Forms for Linear Hamiltonian Vector Fields Commuting with the Action of a Compact Lie Group.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, 1993, pp. 235–68, doi:<a href=\"https://doi.org/10.1017/s0305004100071577\">10.1017/s0305004100071577</a>.","apa":"Melbourne, I., &#38; Dellnitz, M. (1993). Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, 235–268. <a href=\"https://doi.org/10.1017/s0305004100071577\">https://doi.org/10.1017/s0305004100071577</a>","ama":"Melbourne I, Dellnitz M. Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. 1993:235-268. doi:<a href=\"https://doi.org/10.1017/s0305004100071577\">10.1017/s0305004100071577</a>","chicago":"Melbourne, Ian, and Michael Dellnitz. “Normal Forms for Linear Hamiltonian Vector Fields Commuting with the Action of a Compact Lie Group.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, 1993, 235–68. <a href=\"https://doi.org/10.1017/s0305004100071577\">https://doi.org/10.1017/s0305004100071577</a>.","ieee":"I. Melbourne and M. Dellnitz, “Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group,” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, pp. 235–268, 1993."}},{"type":"journal_article","publication":"Archive for Rational Mechanics and Analysis","status":"public","_id":"16634","user_id":"15701","department":[{"_id":"101"}],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0003-9527","1432-0673"]},"year":"1993","citation":{"ama":"Melbourne I, Dellnitz M, Golubitsky M. The structure of symmetric attractors. <i>Archive for Rational Mechanics and Analysis</i>. 1993:75-98. doi:<a href=\"https://doi.org/10.1007/bf00386369\">10.1007/bf00386369</a>","ieee":"I. Melbourne, M. Dellnitz, and M. Golubitsky, “The structure of symmetric attractors,” <i>Archive for Rational Mechanics and Analysis</i>, pp. 75–98, 1993.","chicago":"Melbourne, Ian, Michael Dellnitz, and Martin Golubitsky. “The Structure of Symmetric Attractors.” <i>Archive for Rational Mechanics and Analysis</i>, 1993, 75–98. <a href=\"https://doi.org/10.1007/bf00386369\">https://doi.org/10.1007/bf00386369</a>.","short":"I. Melbourne, M. Dellnitz, M. Golubitsky, Archive for Rational Mechanics and Analysis (1993) 75–98.","mla":"Melbourne, Ian, et al. “The Structure of Symmetric Attractors.” <i>Archive for Rational Mechanics and Analysis</i>, 1993, pp. 75–98, doi:<a href=\"https://doi.org/10.1007/bf00386369\">10.1007/bf00386369</a>.","bibtex":"@article{Melbourne_Dellnitz_Golubitsky_1993, title={The structure of symmetric attractors}, DOI={<a href=\"https://doi.org/10.1007/bf00386369\">10.1007/bf00386369</a>}, journal={Archive for Rational Mechanics and Analysis}, author={Melbourne, Ian and Dellnitz, Michael and Golubitsky, Martin}, year={1993}, pages={75–98} }","apa":"Melbourne, I., Dellnitz, M., &#38; Golubitsky, M. (1993). The structure of symmetric attractors. <i>Archive for Rational Mechanics and Analysis</i>, 75–98. <a href=\"https://doi.org/10.1007/bf00386369\">https://doi.org/10.1007/bf00386369</a>"},"page":"75-98","date_updated":"2022-01-06T06:52:53Z","date_created":"2020-04-16T08:32:44Z","author":[{"full_name":"Melbourne, Ian","last_name":"Melbourne","first_name":"Ian"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"},{"first_name":"Martin","last_name":"Golubitsky","full_name":"Golubitsky, Martin"}],"title":"The structure of symmetric attractors","doi":"10.1007/bf00386369"},{"status":"public","publication":"Lectures in Applied Mathematics","type":"journal_article","language":[{"iso":"eng"}],"_id":"17013","department":[{"_id":"101"}],"user_id":"32643","year":"1993","intvolume":"        29","page":"163-169","citation":{"chicago":"Dellnitz, Michael. “The Equivariant Darboux Theorem.” <i>Lectures in Applied Mathematics</i> 29 (1993): 163–69.","ieee":"M. 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Rösler, “A note on property (T) of orthogonal polynomials,” <i>Archiv der Mathematik</i>, vol. 60, no. 5, pp. 459–463, 1993, doi: <a href=\"https://doi.org/10.1007/bf01202312\">10.1007/bf01202312</a>.","chicago":"Lasser, R., and Margit Rösler. “A Note on Property (T) of Orthogonal Polynomials.” <i>Archiv Der Mathematik</i> 60, no. 5 (1993): 459–63. <a href=\"https://doi.org/10.1007/bf01202312\">https://doi.org/10.1007/bf01202312</a>.","ama":"Lasser R, Rösler M. 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Mechanisms of Symmetry Creation. In: <i>Bifurcation and Symmetry</i>. Basel; 1992. doi:<a href=\"https://doi.org/10.1007/978-3-0348-7536-3_9\">10.1007/978-3-0348-7536-3_9</a>","chicago":"Dellnitz, Michael, Martin Golubitsky, and Ian Melbourne. “Mechanisms of Symmetry Creation.” In <i>Bifurcation and Symmetry</i>. Basel, 1992. <a href=\"https://doi.org/10.1007/978-3-0348-7536-3_9\">https://doi.org/10.1007/978-3-0348-7536-3_9</a>.","ieee":"M. Dellnitz, M. Golubitsky, and I. Melbourne, “Mechanisms of Symmetry Creation,” in <i>Bifurcation and Symmetry</i>, Basel, 1992.","bibtex":"@inbook{Dellnitz_Golubitsky_Melbourne_1992, place={Basel}, title={Mechanisms of Symmetry Creation}, DOI={<a href=\"https://doi.org/10.1007/978-3-0348-7536-3_9\">10.1007/978-3-0348-7536-3_9</a>}, booktitle={Bifurcation and Symmetry}, author={Dellnitz, Michael and Golubitsky, Martin and Melbourne, Ian}, year={1992} }","short":"M. Dellnitz, M. Golubitsky, I. Melbourne, in: Bifurcation and Symmetry, Basel, 1992.","mla":"Dellnitz, Michael, et al. “Mechanisms of Symmetry Creation.” <i>Bifurcation and Symmetry</i>, 1992, doi:<a href=\"https://doi.org/10.1007/978-3-0348-7536-3_9\">10.1007/978-3-0348-7536-3_9</a>.","apa":"Dellnitz, M., Golubitsky, M., &#38; Melbourne, I. (1992). Mechanisms of Symmetry Creation. In <i>Bifurcation and Symmetry</i>. 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Generic Bifurcations of Pendula. In: <i>Bifurcation and Symmetry</i>. Basel; 1992. doi:<a href=\"https://doi.org/10.1007/978-3-0348-7536-3_10\">10.1007/978-3-0348-7536-3_10</a>","chicago":"Dellnitz, Michael, Jerrold E. Marsden, Ian Melbourne, and Jürgen Scheurle. “Generic Bifurcations of Pendula.” In <i>Bifurcation and Symmetry</i>. Basel, 1992. <a href=\"https://doi.org/10.1007/978-3-0348-7536-3_10\">https://doi.org/10.1007/978-3-0348-7536-3_10</a>.","ieee":"M. Dellnitz, J. E. Marsden, I. Melbourne, and J. Scheurle, “Generic Bifurcations of Pendula,” in <i>Bifurcation and Symmetry</i>, Basel, 1992.","apa":"Dellnitz, M., Marsden, J. E., Melbourne, I., &#38; Scheurle, J. (1992). Generic Bifurcations of Pendula. In <i>Bifurcation and Symmetry</i>. 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Dellnitz, I. Melbourne, and J. E. Marsden, “Generic bifurcation of Hamiltonian vector fields with symmetry,” <i>Nonlinearity</i>, pp. 979–996, 1992.","ama":"Dellnitz M, Melbourne I, Marsden JE. Generic bifurcation of Hamiltonian vector fields with symmetry. <i>Nonlinearity</i>. 1992:979-996. doi:<a href=\"https://doi.org/10.1088/0951-7715/5/4/008\">10.1088/0951-7715/5/4/008</a>","short":"M. Dellnitz, I. Melbourne, J.E. Marsden, Nonlinearity (1992) 979–996.","bibtex":"@article{Dellnitz_Melbourne_Marsden_1992, title={Generic bifurcation of Hamiltonian vector fields with symmetry}, DOI={<a href=\"https://doi.org/10.1088/0951-7715/5/4/008\">10.1088/0951-7715/5/4/008</a>}, journal={Nonlinearity}, author={Dellnitz, M and Melbourne, I and Marsden, J E}, year={1992}, pages={979–996} }","mla":"Dellnitz, M., et al. “Generic Bifurcation of Hamiltonian Vector Fields with Symmetry.” <i>Nonlinearity</i>, 1992, pp. 979–96, doi:<a href=\"https://doi.org/10.1088/0951-7715/5/4/008\">10.1088/0951-7715/5/4/008</a>.","apa":"Dellnitz, M., Melbourne, I., &#38; Marsden, J. E. (1992). Generic bifurcation of Hamiltonian vector fields with symmetry. <i>Nonlinearity</i>, 979–996. <a href=\"https://doi.org/10.1088/0951-7715/5/4/008\">https://doi.org/10.1088/0951-7715/5/4/008</a>"},"publication_identifier":{"issn":["0951-7715","1361-6544"]},"publication_status":"published","title":"Generic bifurcation of Hamiltonian vector fields with symmetry","doi":"10.1088/0951-7715/5/4/008","date_updated":"2022-01-06T06:52:52Z","author":[{"full_name":"Dellnitz, M","last_name":"Dellnitz","first_name":"M"},{"first_name":"I","last_name":"Melbourne","full_name":"Melbourne, I"},{"last_name":"Marsden","full_name":"Marsden, J E","first_name":"J E"}],"date_created":"2020-04-15T08:59:25Z"},{"type":"journal_article","publication":"IMA Journal of Numerical Analysis","status":"public","user_id":"32643","department":[{"_id":"101"}],"_id":"17012","language":[{"iso":"eng"}],"issue":"3","publication_status":"published","citation":{"ama":"Dellnitz M. Computational bifurcation of periodic solutions in systems with symmetry. <i>IMA Journal of Numerical Analysis</i>. 1992;12(3):429-455. doi:<a href=\"https://doi.org/10.1093/imanum/12.3.429\">10.1093/imanum/12.3.429</a>","chicago":"Dellnitz, Michael. “Computational Bifurcation of Periodic Solutions in Systems with Symmetry.” <i>IMA Journal of Numerical Analysis</i> 12, no. 3 (1992): 429–55. <a href=\"https://doi.org/10.1093/imanum/12.3.429\">https://doi.org/10.1093/imanum/12.3.429</a>.","ieee":"M. Dellnitz, “Computational bifurcation of periodic solutions in systems with symmetry,” <i>IMA Journal of Numerical Analysis</i>, vol. 12, no. 3, pp. 429–455, 1992.","mla":"Dellnitz, Michael. “Computational Bifurcation of Periodic Solutions in Systems with Symmetry.” <i>IMA Journal of Numerical Analysis</i>, vol. 12, no. 3, 1992, pp. 429–55, doi:<a href=\"https://doi.org/10.1093/imanum/12.3.429\">10.1093/imanum/12.3.429</a>.","short":"M. 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