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 ⁡ 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.

Planar Hamiltonian systems: index theory and applications to the existence of subharmonics

“…5], where the functions b(t) and d(t) are assumed to be, instead, strictly positive. For some sharp estimates of the minimal order k * of these subharmonic solutions, depending on the coefficient supports, we refer to the recent papers [25,26,27].…”