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

Abstract

In this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form x˙=−y+εP1(x,y,z,u,v)+h1(t),y˙=x+εP2(x,y,z,u,v)+h2(t),z˙=−u+εP3(x,y,z,u,v)+h3(t),u˙=z+εP4(x,y,z,u,v)+h4(t),v˙=λv+εP5(x,y,z,u,v)+h5(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 P1,P2,P3,P4{P}_{1},{P}_{2},{P}_{3},{P}_{4}, and P5{P}_{5} are polynomials in the variables x,y,z,u,vx,y,z,u,v of degree nn, hi(t){h}_{i}\left(t) are 2π2\pi -periodic functions with i=1,5¯i=\overline{1,5}, λ\lambda is a real number, and ε\varepsilon is a small parameter.

Keywords

• periodic orbit
• differential system
• averaging theory
• 34c25
• 34c29
• 37g15