Phase Portraits of a Family of Hamiltonian Cubic Systems

Márcio R. A. Gouveia; Jaume Llibre; Luci Any Roberto

doi:10.1007/s12591-024-00691-0

Phase Portraits of a Family of Hamiltonian Cubic Systems

Márcio R. A. Gouveia, Jaume Llibre, Luci Any Roberto

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Resum

While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincaré disc were classified in 1994 (see Artés and Llibre (J Differ Equ 107: 80-95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincaré disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems (Formula presented.) where (x, y) ∈ R2 are the variables and μ is a real parameter. We classify in the Poincaré disc the topological phase portraits of this family of systems when the parameter μ varies, describing the bifurcations which take place.

Idioma original	Anglès
Pàgines (de-a)	747-755
Nombre de pàgines	9
Revista	Differential Equations and Dynamical Systems
Volum	33
Número	2
DOIs	https://doi.org/10.1007/s12591-024-00691-0
Estat de la publicació	Publicada - d’abr. 2025

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10.1007/s12591-024-00691-0

https://ddd.uab.cat/record/312569

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STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY
Llibre Salo, J. (PI), Torregrosa i Arús, J. (Co-Investigador/a Principal), Cufi Cabre, C. (Col.laborador/a), Carvalho, Y. R. (Col.laborador/a), Cousillas, G. (Col.laborador/a), Diz Pita, É. (Col.laborador/a), Ferragut Amengual, A. M. (Col.laborador/a), GOUVEIA, L. F. D. S. (Col.laborador/a), Iván Sánchez Sánchez (Col.laborador/a), Jiménez Ruíz, J. J. (Col.laborador/a), RODERO, A. L. (Col.laborador/a), Ramírez, V. (Col.laborador/a), Artes Ferragud, J. C. (Investigador/a), Caubergh , M. M. (Investigador/a), Cima Mollet, A. M. (Investigador/a), Corbera Subirana, M. (Investigador/a), Cors Iglesias, J. M. (Investigador/a), Gasull Embid, A. (Investigador/a), Pantazi ., C. (Investigador/a), Soler Villanueva, J. (Investigador/a), Garrido Pelaez, J. M. (Col.laborador/a) & Valls Anglés, C. (Col.laborador/a)
1/06/20 → 29/02/24
Projecte: Projectes i Ajuts a la Recerca

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title = "Phase Portraits of a Family of Hamiltonian Cubic Systems",

abstract = "While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincar{\'e} disc were classified in 1994 (see Art{\'e}s and Llibre (J Differ Equ 107: 80-95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincar{\'e} disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems (Formula presented.) where (x, y) ∈ R2 are the variables and μ is a real parameter. We classify in the Poincar{\'e} disc the topological phase portraits of this family of systems when the parameter μ varies, describing the bifurcations which take place.",

keywords = "Cubic polynomial differential systems, Cubic systems, Hamiltonian cubic systems, Poincar{\'e} compactification, Topological phase portraits, Bifurcations",

author = "Gouveia, {M{\'a}rcio R. A.} and Jaume Llibre and Roberto, {Luci Any}",

note = "Publisher Copyright: {\textcopyright} Foundation for Scientific Research and Technological Innovation 2024.",

year = "2025",

month = apr,

doi = "10.1007/s12591-024-00691-0",

language = "English",

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T1 - Phase Portraits of a Family of Hamiltonian Cubic Systems

AU - Gouveia, Márcio R. A.

AU - Llibre, Jaume

AU - Roberto, Luci Any

N1 - Publisher Copyright: © Foundation for Scientific Research and Technological Innovation 2024.

PY - 2025/4

Y1 - 2025/4

N2 - While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincaré disc were classified in 1994 (see Artés and Llibre (J Differ Equ 107: 80-95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincaré disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems (Formula presented.) where (x, y) ∈ R2 are the variables and μ is a real parameter. We classify in the Poincaré disc the topological phase portraits of this family of systems when the parameter μ varies, describing the bifurcations which take place.

AB - While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincaré disc were classified in 1994 (see Artés and Llibre (J Differ Equ 107: 80-95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincaré disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems (Formula presented.) where (x, y) ∈ R2 are the variables and μ is a real parameter. We classify in the Poincaré disc the topological phase portraits of this family of systems when the parameter μ varies, describing the bifurcations which take place.

KW - Cubic polynomial differential systems

KW - Cubic systems

KW - Hamiltonian cubic systems

KW - Poincaré compactification

KW - Topological phase portraits

KW - Bifurcations

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JO - Differential Equations and Dynamical Systems

JF - Differential Equations and Dynamical Systems

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Phase Portraits of a Family of Hamiltonian Cubic Systems

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STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY

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