Affiliation:
1. Department of Mathematics, Laboratory LMA, University of Annaba , El Hadjar , 23000 Annaba , Algeria
Abstract
Abstract
In this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form
x
˙
=
−
y
+
ε
P
1
(
x
,
y
,
z
,
u
,
v
)
+
h
1
(
t
)
,
y
˙
=
x
+
ε
P
2
(
x
,
y
,
z
,
u
,
v
)
+
h
2
(
t
)
,
z
˙
=
−
u
+
ε
P
3
(
x
,
y
,
z
,
u
,
v
)
+
h
3
(
t
)
,
u
˙
=
z
+
ε
P
4
(
x
,
y
,
z
,
u
,
v
)
+
h
4
(
t
)
,
v
˙
=
λ
v
+
ε
P
5
(
x
,
y
,
z
,
u
,
v
)
+
h
5
(
t
)
,
\begin{array}{r}\dot{x}=-y+\varepsilon {P}_{1}\left(x,y,z,u,v)+{h}_{1}\left(t),\\ \dot{y}=x+\varepsilon {P}_{2}\left(x,y,z,u,v)+{h}_{2}\left(t),\\ \dot{z}=-u+\varepsilon {P}_{3}\left(x,y,z,u,v)+{h}_{3}\left(t),\\ \dot{u}=z+\varepsilon {P}_{4}\left(x,y,z,u,v)+{h}_{4}\left(t),\\ \dot{v}=\lambda v+\varepsilon {P}_{5}\left(x,y,z,u,v)+{h}_{5}\left(t),\end{array}
where
P
1
,
P
2
,
P
3
,
P
4
{P}_{1},{P}_{2},{P}_{3},{P}_{4}
, and
P
5
{P}_{5}
are polynomials in the variables
x
,
y
,
z
,
u
,
v
x,y,z,u,v
of degree
n
n
,
h
i
(
t
)
{h}_{i}\left(t)
are
2
π
2\pi
-periodic functions with
i
=
1
,
5
¯
i=\overline{1,5}
,
λ
\lambda
is a real number, and
ε
\varepsilon
is a small parameter.
Subject
Applied Mathematics,Numerical Analysis,Statistics and Probability,Analysis