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

Abstract

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⁡βZ:=\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.

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
• 34c25
• 37b55
• 37e40