Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory

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.