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The goal of this article is to study the existence of closed trajectories forthe differential equation $\dddot{z}+a\ddot{z}+b\dot{z}+abz=\varepsilonF(z,\dot{z},\ddot{z})$ in two situations. By considering$F(z,\dot{z},\ddot{z})=1$ and $b=sgn(h(z,\dot{z},\ddot{z}))$, where$h(z,\dot{z},\ddot{z})=z^2+(\dot{z})^2+(\ddot{z})^2-1$, the differentialequation is equivalent to the piecewise smooth differential system that admitsthe unit sphere as the discontinuity manifold. In this case we obtainconditions for the existence of a closed pseudo-trajectory. Considering$\varepsilon \neq 0$ sufficiently small, $b>0$ and $F(z,\dot{z},\ddot{z})$ a$n$-degree polynomial, we show that the unperturbed differential equation has afamily of isochronous periodic solutions filling an invariant plane and studythe maximum number of limit cycles which bifurcate from this $2$-dimensionalisochronous using the averaging theory.

On the existence of closed trajectories for the differential equation $\dddot{z}+a\ddot{z}+b\dot{z}+abz=\varepsilon F(z,\dot{z},\ddot{z})$