Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families
This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, st...
- Autores:
-
Rodríguez Contreras, Jorge
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad del Atlántico
- Repositorio:
- Repositorio Uniatlantico
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniatlantico.edu.co:20.500.12834/1143
- Acceso en línea:
- https://hdl.handle.net/20.500.12834/1143
- Palabra clave:
- Quadratic Polynomial Systems, Critical Points, Bifurcations, Stable Manifold, Phase portraits of polynomial systems.
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc/4.0/
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Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| dc.title.alternative.spa.fl_str_mv |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| title |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| spellingShingle |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families Quadratic Polynomial Systems, Critical Points, Bifurcations, Stable Manifold, Phase portraits of polynomial systems. |
| title_short |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| title_full |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| title_fullStr |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| title_full_unstemmed |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| title_sort |
Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families |
| dc.creator.fl_str_mv |
Rodríguez Contreras, Jorge |
| dc.contributor.author.none.fl_str_mv |
Rodríguez Contreras, Jorge |
| dc.contributor.other.none.fl_str_mv |
Reyes Linero, Alberto Blanco Montes, Bladimir B. Acosta-Humánez, Primitivo |
| dc.subject.keywords.spa.fl_str_mv |
Quadratic Polynomial Systems, Critical Points, Bifurcations, Stable Manifold, Phase portraits of polynomial systems. |
| topic |
Quadratic Polynomial Systems, Critical Points, Bifurcations, Stable Manifold, Phase portraits of polynomial systems. |
| description |
This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis of these families are graphed. To properly perform this study it was necessary to use some results of the non-linear systems theory, for this reason vital definitions and theorems were included because of their importance during the study of the multiparametric families. Algebraic aspects are also included. |
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2021 |
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2021-01-05 |
| dc.date.submitted.none.fl_str_mv |
2021-04-17 |
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2022-12-19T02:42:38Z |
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2022-12-19T02:42:38Z |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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Artículo |
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publishedVersion |
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Contreras, J. R., Linero, A. R., Montes, B. B., & Acosta-Humánez, P. B. (2021). Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families. arXiv preprint arXiv:2103.02773. |
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https://hdl.handle.net/20.500.12834/1143 |
| dc.identifier.doi.none.fl_str_mv |
10.37394/23206.2021.20.20 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad del Atlántico |
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Repositorio Universidad del Atlántico |
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Contreras, J. R., Linero, A. R., Montes, B. B., & Acosta-Humánez, P. B. (2021). Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families. arXiv preprint arXiv:2103.02773. 10.37394/23206.2021.20.20 Universidad del Atlántico Repositorio Universidad del Atlántico |
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https://hdl.handle.net/20.500.12834/1143 |
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eng |
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eng |
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Attribution-NonCommercial 4.0 International |
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info:eu-repo/semantics/openAccess |
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http://creativecommons.org/licenses/by-nc/4.0/ Attribution-NonCommercial 4.0 International http://purl.org/coar/access_right/c_abf2 |
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Sede Norte |
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World Scientific and Engineering Academy and Society |
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Rodríguez Contreras, Jorgec3f6fcbd-e002-4b97-ab75-4cdce01e825fReyes Linero, AlbertoBlanco Montes, BladimirB. Acosta-Humánez, Primitivo2022-12-19T02:42:38Z2022-12-19T02:42:38Z2021-01-052021-04-17Contreras, J. R., Linero, A. R., Montes, B. B., & Acosta-Humánez, P. B. (2021). Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families. arXiv preprint arXiv:2103.02773.https://hdl.handle.net/20.500.12834/114310.37394/23206.2021.20.20Universidad del AtlánticoRepositorio Universidad del AtlánticoThis article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis of these families are graphed. To properly perform this study it was necessary to use some results of the non-linear systems theory, for this reason vital definitions and theorems were included because of their importance during the study of the multiparametric families. Algebraic aspects are also included.application/pdfenghttp://creativecommons.org/licenses/by-nc/4.0/Attribution-NonCommercial 4.0 Internationalinfo:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2World Scientific and Engineering Academy and SocietyTranscritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric FamiliesTranscritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric FamiliesPúblico generalQuadratic Polynomial Systems, Critical Points, Bifurcations, Stable Manifold, Phase portraits of polynomial systems.info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArtículohttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_2df8fbb1BarranquillaMatemáticasSede NorteAcosta-Humanez P.B., Lazaro J.T., Morales-Ruiz J.J. & Pantazi Ch. ´ , On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Discrete & Continuous Dynamical Systems-A 35 (2015): 1767–1800.. Available at arXiv:1012.4796.Acosta-Humanez P. B., Reyes Linero A. & Rodr ´ ´ıguez Contreras J., Algebraic and qualitative remarks about the family yy0 = (αxm+k−1 + βxm−k−1 )y + γx2m−2k−1 , preprint 2014. Available at arXiv:1807.03551.Rodr´ıguez Contreras J., Acosta-Humanez P. B. & Reyes Linero A. ´ , Algebraic and qualitative remarks about the family yy0 = (αxm+k−1 + βxm−k−1 )y + γx2m−2k−1 , Open Mathematics 17 (2019), 1220–1238.Acosta-Humanez P. B., Reyes Linero A. & Rodr ´ ´ıguez Contreras J.,Galoisian and Qualitative Approaches to Linear Polyanin-Zaitsev Vector Fields, Open Mathematics 16 (2018), 1204–1217. Available at arXiv:1807.05272.Acosta-Hum´anez P. B., Campo Donado M., Reyes Linero A., & Rodr´ıguez Contreras J., (2019). Algebraic and qualitative aspects of quadratic vector fields related with classical orthogonal polynomials. arXiv preprint arXiv:1906.09764.Rodr´ıguez Contreras J, Reyes Linero A., Campo Donado M., & Acosta-Hum´anez P. B. , (2020). Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials. Journal of Southwest Jiaotong University, 55(4).Torres Henao J. A. , Sistemas Din´amicos Planos. Universidad Nacional de Colombia, Facultad de Ciencias, Escuela de Matem´aticas, Medellin, Colombia (2013). http://bdigital.unal.edu.co/9478/V´ılchez Lobato M. L. , Velasco Morente F., Garc´ıa del Hoyo J. J.,Bifurcaciones transcr´ıticas y ciclos l´ımites en un modelo din´amico de competici´on entre dos especies. Una aplicaci´on a la pescadera de engraulis encrasicholus de la Regi´on Suratl´antica espan˜nola, (2002).Xiangdong, X., & Jianfeng, Z.,Plane Polynomial System and it’s Accompany System,Journal of Mathematical Study, 37(2) (2004), 161–166.Gaiko V.A., Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model, Nonlinear Analysis: Theory, Methods & Applications, 74(18), (2011) 7532–7542Perko,L., Differential Equations and Dynamical Systems (New York: Springer Verlag), 2001] Guckeinheimer, J., y Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields ( New York: Springer Verlag), 1983Dumortier F., Llibre J. & Artes J.,Qualitative Theory Of Planar Differential Systems, (Berlin: Springer), 2006Hale J.K & Koc¸ak H., Dynamics and Bifurcations, (Springer-Verlag), 1991.Herssens C., Maesschalck P., Artes J. C., Dumortier F. & Llibre J. ´ , P4 (http://mat.uab.es/ artes/p4/p4.htm), Dept. de Matem`atiques, Universitat Aut`onoma de Barcelona.Computational algebraic system, dynamic greometry software; GeogebraComputational algebraic system, dynamic greometry software; GeogebraAcosta-Hum´anez, M. F., Acosta-Hum´anez, P. B., & Tuir´an, Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 14, (2018) 099.Acosta-Humanez, P. B., Blazquez-Sanz, D., & Vargas-Contreras, C. A. On Hamiltonian potentials with quartic polynomial normal variational equations, Nonlinear Studies, 16(3), (2009) 299–314.Acosta-Hum´anez, P., & Bl´azquez-Sanz, D. Non-integrability of some hamiltonians with rational potentials, Discrete & Continuous Dynamical Systems-B, 10(2&3), (2008) 265–293.Acosta-Hum´anez, P. B. La teor´ıa de Morales-Ramis y el algoritmo de Kovacic. Lecturas Matem´aticas, 27(3), (2006) 21–56.Acosta-Hum´anez, P. B. Nonautonomous Hamiltonian Systems and Morales–Ramis Theory I. The case x¨ = f(x, t), SIAM Journal on Applied Dynamical Systems, 8(1), (2009) 279–297.Acosta-Hum´anez, P. B., & Pantazi, C. Darboux integrals for Schr¨odinger planar vector fields via Darboux transformations, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 8 (2012) 043.Acosta-Hum´anez, P. B., Alvarez-Ram´ırez, M.,& Delgado, J. ´ Non-integrability of some few body problems in two degrees of freedom, Qualitative theory of dynamical systems, 8(2), (2009) 209–239.Acosta-Hum´anez, P. B., Alvarez-Ram´ırez, M., Bl´azquez-Sanz, D., & Delgado, J. Nonintegrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum, Discrete & Continuous Dynamical Systems-A, 33(3), (2013) 965–986.Acosta-Hum´anez, P. B., L´azaro, J. T., Morales-Ruiz, J. J., & Pantazi, C. Differential Galois theory and non-integrability of planar polynomial vector fields, Journal of Differential Equations, 264(12), 7183–7212.Acosta-Hum´anez, P. B., & Bl´azquez-Sanz, D. A. V. I. D. Hamiltonian system and variational equations with polynomial coefficients Dynamic systems and applications, Dynamic, Atlanta, GA, 5, (2008) 6–10.Acosta-Hum´anez, P. B., Bl´azquez-Sanz, D., & Venegas-G´omez, H. Liouvillian solutions for second order linear differential equations with polynomial coefficients, S˜ao Paulo Journal of Mathematical Sciences, (2020) 1–20.Acosta-Hum´anez, P. B. M´etodos algebraicos en sistemas din´amicos, Ediciones Universidad del Atl´antico, EMALCA 2014.Acosta-Hum´anez, P. B., & Yagasaki, K. Nonintegrability of the unfoldings of codimension-two bifurcations, Nonlinearity, 33(4), (2020) 1366–1387.Abramowitz, M. and Stegun, I. Handbook of Mathematical Functions, ninth printing, New York: Dover (1972).http://purl.org/coar/resource_type/c_2df8fbb1ORIGINALTranscritical_Bifurcations_and_Algebraic_Aspects_o.pdfTranscritical_Bifurcations_and_Algebraic_Aspects_o.pdfapplication/pdf5039758https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/1143/1/Transcritical_Bifurcations_and_Algebraic_Aspects_o.pdf5f9886b3cee2080a2e4d956ec4197748MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8914https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/1143/2/license_rdf24013099e9e6abb1575dc6ce0855efd5MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81306https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/1143/3/license.txt67e239713705720ef0b79c50b2ececcaMD5320.500.12834/1143oai:repositorio.uniatlantico.edu.co:20.500.12834/11432022-12-18 21:42:39.141DSpace de la Universidad de Atlánticosysadmin@mail.uniatlantico.edu.coVMOpcm1pbm9zIGdlbmVyYWxlcyBkZWwgUmVwb3NpdG9yaW8gSW5zdGl0dWNpb25hbCBkZSBsYSBVbml2ZXJzaWRhZCBkZWwgQXRsw6FudGljbwoKRWwgKGxvcykgYXV0b3IgKGVzKSBoYW4gYXNlZ3VyYWRvIChuKSBsbyBzaWd1aWVudGUgc29icmUgbGEgb2JyYSBhIGludGVncmFyIGVuIGVsIFJlcG9zaXRvcmlvIEluc3RpdHVjaW9uYWwsIHF1ZToKCuKXjwlFcyBvcmlnaW5hbCwgZGUgc3UgZXhjbHVzaXZhIGF1dG9yw61hLCBzZSByZWFsaXrDsyBzaW4gdmlvbGFyIG8gdXN1cnBhciBkZXJlY2hvcyBkZSBhdXRvciBkZSB0ZXJjZXJvcyB5IHBvc2VlIGxhIHRpdHVsYXJpZGFkLgril48JQXN1bWlyw6FuIGxhIHJlc3BvbnNhYmlsaWRhZCB0b3RhbCBwb3IgZWwgY29udGVuaWRvIGEgbGEgb2JyYSBhbnRlIGxhIEluc3RpdHVjacOzbiB5IHRlcmNlcm9zLgril48JQXV0b3JpemFuIGEgdMOtdHVsbyBncmF0dWl0byB5IHJlbnVuY2lhcyBhIHJlY2liaXIgZW1vbHVtZW50b3MgcG9yIGxhcyBhY3RpdmlkYWRlcyBxdWUgc2UgcmVhbGljZW4gY29uIGVsbGEsIHNlZ8O6biBzdSBsaWNlbmNpYS4KCgpMYSBVbml2ZXJzaWRhZCBkZWwgQXRsw6FudGljbywgcG9yIHN1IHBhcnRlLCBzZSBjb21wcm9tZXRlIGEgYWN0dWFyIGVuIGxvcyB0w6lybWlub3MgZXN0YWJsZWNpZG9zIGVuIGxhIExleSAyMyBkZSAxOTgyIHkgbGEgRGVjaXNpw7NuIEFuZGluYSAzNTEgZGUgMTk5MywgZGVtw6FzIG5vcm1hcyBnZW5lcmFsZXMgc29icmUgbGEgbWF0ZXJpYSB5IGVsIEFjdWVyZG8gU3VwZXJpb3IgMDAxIGRlIDE3IGRlIG1hcnpvIGRlIDIwMTEsIHBvciBtZWRpbyBkZWwgY3VhbCBzZSBleHBpZGUgZWwgRXN0YXR1dG8gZGUgUHJvcGllZGFkIEludGVsZWN0dWFsIGRlIGxhIFVuaXZlcnNpZGFkIGRlbCBBdGzDoW50aWNvLgoKUG9yIMO6bHRpbW8sIGhhbiBzaWRvIGluZm9ybWFkb3Mgc29icmUgZWwgdHJhdGFtaWVudG8gZGUgZGF0b3MgcGVyc29uYWxlcyBwYXJhIGZpbmVzIGFjYWTDqW1pY29zIHkgZW4gYXBsaWNhY2nDs24gZGUgY29udmVuaW9zIGNvbiB0ZXJjZXJvcyBvIHNlcnZpY2lvcyBjb25leG9zIGNvbiBhY3RpdmlkYWRlcyBwcm9waWFzIGRlIGxhIGFjYWRlbWlhLCBiYWpvIGVsIGVzdHJpY3RvIGN1bXBsaW1pZW50byBkZSBsb3MgcHJpbmNpcGlvcyBkZSBsZXkuCgpMYXMgY29uc3VsdGFzLCBjb3JyZWNjaW9uZXMgeSBzdXByZXNpb25lcyBkZSBkYXRvcyBwZXJzb25hbGVzIHB1ZWRlbiBwcmVzZW50YXJzZSBhbCBjb3JyZW8gZWxlY3Ryw7NuaWNvIGhhYmVhc2RhdGFAbWFpbC51bmlhdGxhbnRpY28uZWR1LmNvCg== |