On the existence of closed trajectories and pseudo-trajectories for a family of third order differential equations

Abstract

The goal of this article is to study the existence of closed trajectories for the differential equation $\stackrel{...}{z}+a\ddot{z}+b\dot{z}+abz=\epsilon F(z,\dot{z},\ddot{z})$ in two situations. In the first situation, we consider $F(z,\dot{z},\ddot{z})=1$ and $b=\mathrm{sgn}(h(z,\dot{z},\ddot{z}))$, 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 $\epsilon \ne 0$ 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).

25 pages, 1 figure