DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

UNBOUNDEDNESS OF SOLUTIONS OF PLANAR
HAMILTONIAN SYSTEMS
XIAOJING YANG
The unboundedness of solutions for the following planar Hamiltonian system: Ju ′ =
∇H(u)+h(t) is discussed, where the function H(u) ∈ C 3 (R 2 ,R)a.e.inR 2 , is positive for
u ≠ 0 and positively homogeneous of degree 2, h ∈ L 1 [0,2π] × L 1 [0,2π] is2π-periodic,
and J is the standard symplectic matrix.
Copyright © 2006 Xiaojing Yang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
We are interested in this paper in the unboundedness of solutions of the planar Hamiltonian
system
Ju ′ =∇H(u)+h(t),
(
′ = d )
, (1.1)
dt
where H(u) ∈ C 3 (R 2 ,R) a.e.inR 2 is positive for u ≠ 0, and positively homogeneous of
degree 2, that is, for every u ∈ R 2 , λ>0,
H(λu) = λ 2 H(u),
min H(u) > 0, (1.2)
‖u‖=1
h ∈ L 1 [0,2π] × L 1 [0,2π]is2π-periodic, and J = ( 0 −1
1 0 ) is the standard symplectic matrix.
Under the above conditions, it is easy to see that the origin is an isochronous center
for the autonomous system
Ju ′ =∇H(u), (1.3)
that is, all solutions of (1.3) are periodic with the same minimal period, which will be
denoted as τ.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1167–1176