The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems

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López Gómez, Julián and Muñoz Hernández, Eduardo and Zanolin, Fabio (2021) The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems. Advanced Nonlinear Studies, 21 (3). pp. 489-499. ISSN 1536-1365

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Official URL: https://doi.org/10.1515/ans-2021-2137




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.


Item Type:Article
Uncontrolled Keywords:Periodic Predator-Prey Model of Volterra Type; Subharmonic Coexistence States; Poincaré–Birkhoff Twist Theorem; Degenerate Versus Non Degenerate Models; Point-Wise Behavior of the Low-Order Subharmonics as the Model Degenerates
Subjects:Sciences > Mathematics > Differential equations
Sciences > Mathematics > Topology
ID Code:74336
Deposited On:05 Sep 2022 16:44
Last Modified:06 Sep 2022 07:38

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