Periodic solutions for differential systems in ℝ³ and ℝ⁴

We provide sufficient conditions for the existence of periodic solutions for the differential systems x′=y, y′=z, z′=−y−εF(t,x,y,z), andx′=y, y′=−x−εG(t,x,y,z,u), z′=u, u′=−z−εH(t,x,y,z,u), \matrix{{x' = y,\;\;\;y' = z,\;\;\;z' = - y - \varepsilon F(t,x,y,z),\;\;\;{\rm{and}}} \cr {x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u),} \hfill \cr } where F, G and H are 2π–periodic functions in the variable t and ɛ is a small parameter. These differential systems appear frequently in many problems coming from the sciences and the engineering.