Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem

Abstract

In the general setting of a planar first order system (0.1) u ′ = G ( t , u ) , u ∈ R 2 , with G : [ 0 , T ] × R 2 → R 2 , we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G ( t , 0 ) ≡ 0 , at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T -periodic solutions of unforced planar Hamiltonian systems J u ′ = ∇ u H ( t , u ) and unforced undamped scalar second order equations x ″ + g ( t , x ) = 0 . In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.