Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

Abstract

We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian

“…(1). Recently, Boscaggin, Garrione and Feltrin, Fonda and Toader, Donde and Zanolin applied the Poincaré-Birkhoff twist theorem to discuss related problems, see [8,7,19,14]. This paper is another result in this direction.…”

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

“…Finally, we recall that a different kind of multiplicity results for periodic Hamiltonian systems is contained in [5,6].…”

On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models

“…Notice that the above proof also ensures that all the solutions are globally defined, as anticipated in Remark 1. By (14) and (16) the twist condition holds and hence, according to Theorem 2.1, the system (6) has, at least, two mT -periodic solutions with rotation number k, for each λ > λ k m . This ends the proof.…”

“…Due to the Hamiltonian structure of the new system, a powerful tool to prove the existence of nontrivial periodic solutions is the Poincaré-Birkhoff twist fixed point theorem. Applications of this method to Volterra predator-prey systems can be found in [18], [10], [11], [26], [2], as well as in the recent ones [16] and [13]. A typical strategy of proof consists in showing that there is a sufficiently large gap in the rotation numbers between the small solutions and the large solutions of (4), which, in turns, guarantees a suitable twist property for the associated Poincaré map.…”

“…In this context, increasing the value of the parameter λ > 0 in (1) may play the role of enlarging the gap in the rotation numbers. If we try to apply the above quoted results [10,11,13,16,18,26] to (1), we find that at least one of the two coefficients α(t), β(t) should be assumed to be always strictly positive and this prevents the applications to models where the kinetics may vanish in some time interval. On the other hand, for the special choice of α(t) and β(t) shown in Figure 1, the results of [19] establish that (1) does not admit a nontrivial T -periodic coexistence state, independently of the value of λ, and therefore it becomes apparent that the Poincaré-Birkhoff theorem cannot be applied to (1) for some specific coefficients like those in Figure 1.…”