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

TL;DR

This paper investigates the existence of closed and pseudo-closed trajectories in a third-order differential equation, using piecewise analysis and averaging theory to identify conditions for periodic and limit cycle solutions.

Contribution

It introduces new conditions for closed pseudo-trajectory existence and analyzes bifurcations of limit cycles in a piecewise smooth third-order system.

Findings

Conditions for closed pseudo-trajectory existence are established.

Maximum number of bifurcating limit cycles is determined.

Isochronous periodic solutions are characterized in the unperturbed system.

Abstract

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).