Abstract
We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.
Highlights
Let us start by considering a planar Hamiltonian system∂H x = (t, x, y), y = − (t, x, y) . (1) ∂y ∂xThe Hamiltonian function H(t, x, y) is assumed to be continuous, T -periodic in t, and continuously differentiable in x and y
R = ]a1,1, a1,2[ × ]a2,1, a2,2[, with ai,j ∈ R ∪ {−∞, +∞}. This means that R can be either a rectangle or an unbounded set, like a quadrant, a half-plane, or even the whole plane R2
We assume that the Hamiltonian function may be decomposed as
Summary
There exists an integer K0 such that, for every integer K > K0, the Hamiltonian system (1) has at least two T -periodic solutions performing exactly K clockwise rotations around (x0, y0) in the time interval [0, T ]. Every solution z(t) to (7), with p > p2, which remains outside R(p2), has to rotate clockwise, with an angular velocity bounded below by b.
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