The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
| Autor: | López-Gómez Julián, Muñoz-Hernández Eduardo, Zanolin Fabio |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Advanced Nonlinear Studies, Vol 21, Iss 3, Pp 489-499 (2021) |
| Druh dokumentu: | article |
| ISSN: | 1536-1365 2169-0375 |
| DOI: | 10.1515/ans-2021-2137 |
| Popis: | 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)x^{\prime}=-\lambda\alpha(t)f(y), y′=λβ(t)g(x)y^{\prime}=\lambda\beta(t)g(x), where α,β\alpha,\beta are non-negative 𝑇-periodic coefficients and λ>0\lambda>0. We focus our study to the so-called “degenerate” situation, namely when the set Z:=suppα∩suppβZ:=\operatorname{supp}\alpha\cap\operatorname{supp}\beta 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 (as well as subharmonic solutions) is guaranteed (for λ>0\lambda>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. |
| Databáze: | Directory of Open Access Journals |
| Externí odkaz: |
|