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Non-resonance and Double Resonance for a Planar System via Rotation Numbers
We consider a planar system z ′ = f ( t , z ) under non-resonance or double resonance conditions and obtain the existence of 2 π -periodic solutions by combining a rotation number approach together with Poincaré-Bohl theorem. Firstly, we allow that the angular velocity of solutions of z ′ = f ( t ,...
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| Published in: | Resultate der Mathematik 2021-05, Vol.76 (2), Article 91 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We consider a planar system
z
′
=
f
(
t
,
z
)
under non-resonance or double resonance conditions and obtain the existence of
2
π
-periodic solutions by combining a rotation number approach together with Poincaré-Bohl theorem. Firstly, we allow that the angular velocity of solutions of
z
′
=
f
(
t
,
z
)
is controlled by the angular velocity of solutions of two positively homogeneous system
z
′
=
L
i
(
t
,
z
)
,
i
=
1
,
2
, whose rotation numbers satisfy
ρ
(
L
1
)
>
n
and
ρ
(
L
2
)
<
n
+
1
, namely, nonresonance occurs in the sense of the rotation number. Secondly, we prove the existence of
2
π
-periodic solutions when the nonlinearity is allowed to interact with two positively homogeneous system
z
′
=
L
i
(
t
,
z
)
,
i
=
1
,
2
, with
ρ
(
L
1
)
≥
n
and
ρ
(
L
2
)
≤
n
+
1
, which gives rise to double resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. |
|---|---|
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-021-01401-w |