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.