RT Journal Article
T1 The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
A1 López Gómez, Julián
A1 Muñoz Hernández, Eduardo
A1 Zanolin, Fabio
AB 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.
PB De Gruyter
SN 1536-1365
YR 2021
FD 2021-07-17
LK https://hdl.handle.net/20.500.14352/4981
UL https://hdl.handle.net/20.500.14352/4981
LA eng
NO [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 OrdinaryDifferential 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 thePoincaré–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.
NO Ministerio de Ciencia e Innovación (MICINN)
DS Docta Complutense
RD 28 abr 2024