  The goal of this article is to study the existence of closed trajectories for
the differential equation $\dddot{z}+a\ddot{z}+b\dot{z}+abz=\varepsilon
F(z,\dot{z},\ddot{z})$ in two situations. In the first situation, we consider
$F(z,\dot{z},\ddot{z})=1$ and $b={\rm sgn}(h(z,\dot{z},\ddot{z}))$, where
$h(z,\dot{z},\ddot{z})=z^2+(\dot{z})^2+(\ddot{z})^2-1$. We show that the
differential equation is equivalent to a piecewise smooth differential system
that admits the unit sphere as the discontinuity manifold. We obtain conditions
for the existence of a closed pseudo-trajectory in this case. In the second
situation, we consider $\varepsilon \neq 0$ sufficiently small, $b>0$, and
$F(z,\dot{z},\ddot{z})$ a $n$-degree polynomial. We show that the unperturbed
differential equation has a family of isochronous periodic solutions filling an
invariant plane. Then, we study the maximum number of limit cycles which
bifurcate from this 2-dimensional isochronous using the averaging theory. Thus,
within the same family, we have periodic solutions (in the case where the
parameters create a smooth equation) and also pseudo-periodic solutions (in the
case of Filippov systems).
