The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
| dc.contributor.author | López Gómez, Julián | |
| dc.contributor.author | Muñoz Hernández, Eduardo | |
| dc.contributor.author | Zanolin, Fabio | |
| dc.date.accessioned | 2023-06-16T14:25:04Z | |
| dc.date.available | 2023-06-16T14:25:04Z | |
| dc.date.issued | 2021-07-17 | |
| dc.description.abstract | In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x' = −λα(t)f (y), y' = λβ(t)g(x), where α, β are non-negative T-periodic coefficients and λ > 0. We focus our study to the so-called “degenerate” situation, namely when the set Z := supp α ∩ supp β has Lebesgue measure zero. It is known that, in this case, for some choices of α and β, no nontrivial T-periodic solution exists. On the opposite, we show that, depending of some geometric configurations of α and β, the existence of a large number of T-periodic solutions (aswell as subharmonic solutions) is guaranteed (for λ > 0 and large). Our proof is based on the Poincaré–Birkhoff twist theorem. Applications are given to Volterra’s predator-prey model with seasonal effects. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación (MICINN) | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/74336 | |
| dc.identifier.doi | 10.1515/ans-2021-2137 | |
| dc.identifier.issn | 1536-1365 | |
| dc.identifier.officialurl | https://doi.org/10.1515/ans-2021-2137 | |
| dc.identifier.relatedurl | https://www.degruyter.com/document/doi/10.1515/ans-2021-2137/html | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/4981 | |
| dc.issue.number | 3 | |
| dc.journal.title | Advanced Nonlinear Studies | |
| dc.language.iso | eng | |
| dc.page.final | 499 | |
| dc.page.initial | 489 | |
| dc.publisher | De Gruyter | |
| dc.relation.projectID | PGC2018-097104-B-100 | |
| dc.rights | Atribución 3.0 España | |
| dc.rights.accessRights | open access | |
| dc.rights.uri | https://creativecommons.org/licenses/by/3.0/es/ | |
| dc.subject.keyword | Periodic Predator-Prey Model of Volterra Type | |
| dc.subject.keyword | Subharmonic Coexistence States | |
| dc.subject.keyword | Poincaré–Birkhoff Twist Theorem | |
| dc.subject.keyword | Degenerate Versus Non Degenerate Models | |
| dc.subject.keyword | Point-Wise Behavior of the Low-Order Subharmonics as the Model Degenerates | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.ucm | Topología | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.subject.unesco | 1210 Topología | |
| dc.title | The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems | |
| dc.type | journal article | |
| dc.volume.number | 21 | |
| dcterms.references | [1] A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud. 11 (2011), no. 1, 77–103. [2] A. Boscaggin and F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst. 33 (2013), 89–110. [3] F. Dalbono and C. Rebelo, Poincaré–Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Rend. Semin. Mat. Univ. Politec. Torino 60 (2002), 233–264. [4] T. R. Ding and F. Zanolin, Harmonic solutions and subharmonic solutions for periodic Lotka–Volterra systems, in: Dynamical Systems (Tianjin 1990/1991), Nankai Ser. Pure Appl. Math. Theoret. Phys. 4, World Scientific, River Edge (1993), 55–65. [5] T. R. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka–Volterra type, in: World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa 1992), De Gruyter, Berlin (1996), 395–406. [6] W. Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, Acta Math. Sinica 25 (1982), no. 2, 227–235. [7] T. Dondè and F. Zanolin, Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré–Birkhoff approach, Topol. Methods Nonlinear Anal. 55 (2020), no. 2, 565–581. [8] A. Fonda, Playing Around Resonance. An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, Cham, 2016. [9] A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré–Birkhoff theorem, Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 29–52. [10] A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal. 8 (2019), no. 1, 583–602. [11] A. Fonda and A. J. Ureña, A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, 679–698. [12] A. R. Hausrath and R. F. Manásevich, Periodic solutions of a periodically perturbed Lotka–Volterra equation using the Poincaré–Birkhoff theorem, J. Math. Anal. Appl. 157 (1991), no. 1, 1–9. [13] J. López-Gómez, A bridge between operator theory and mathematical biology, in: Operator Theory and its Applications (Winnipeg 1998), Fields Inst. Commun. 25, American Mathematical Society, Providence (2000), 383–397. [14] J. López-Gómez and E. Muñoz Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity 33 (2020), no. 1, 34–71. [15] J. López-Gómez, E. Muñoz Hernández and F. Zanolin, On the applicability of the Poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models, Discrete Contin. Dyn. Syst. 40 (2020), no. 4, 2393–2419. [16] J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka–Volterra model, Adv. Differential Equations 1 (1996), no. 3, 403–423. [17] A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré–Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations 183 (2002), no. 2, 342–367. [18] C. Rebelo, A note on the Poincaré–Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal. 29 (1997), no. 3, 291–311. | |
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