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

TL;DR

It is shown that the unperturbed differential equation has a family of isochronous periodic solutions filling an invariant plane and the maximum number of limit cycles which bifurcate from this $2$-dimensional isochronic using the averaging theory.

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).

Figures (1)

• Figure 1: Vector field $Z_{X_-X_+}$ with $a=5$ and $\varepsilon=4$. The heart-shaped curve in (b) is the pseudo-cycle.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Descartes' Theorem
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3