The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
| Autor: | Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Poincaré-Birkhoff Twist Theorem
Pure mathematics Class (set theory) Periodic Predator-Prey Model of Volterra Type Topología General Mathematics 010102 general mathematics Degenerate energy levels Subharmonic Coexistence States Statistical and Nonlinear Physics 01 natural sciences Point-Wise Behavior of the Low-Order Subharmonics as the Model Degenerates Hamiltonian system Degenerate Versus Non-Degenerate Models 010101 applied mathematics Planar Poincaré–Birkhoff theorem Ecuaciones diferenciales 0101 mathematics Mathematics |
| Zdroj: | E-Prints Complutense. Archivo Institucional de la UCM instname |
| Popis: | In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x ′ = - λ α ( t ) f ( y ) x^{\prime}=-\lambda\alpha(t)f(y) , y ′ = λ β ( 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 := supp α ∩ supp β Z:=\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: | OpenAIRE |
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