We provide sufficient conditions for the existence of periodic solutions for the differential systems \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.

We provide sufficient conditions for the existence of periodic solutions for the class of Duffing differential equations, where the functions a(t), b(t), c(t) and h(t, x) are C 2 and T-periodic in the variable t.

The averaging theory of second order shows that for polynomial differential systems in R 4 with cubic homogeneous nonlinearities at least nine limit cycles can be born in a zero-Hopf bifurcation.

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