Rigid centres on the center manifold of tridimensional differential systems

Abstract

Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on $\mathbb {R}^{3}$ . On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on $\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems.

Cyclicity of Rigid Centers on Center Manifolds of Three-dimensional systems

“…Rigid Systems in R 3 . We exhibit a quick overview of rigid systems in R 3 , which are studied with more detail in [16].…”

“…In the paper [16], the authors extended the concept of rigid systems and rigid centers for three-dimensional systems (1). In essence, system (1) has a rigid center at the origin when its restriction to the center manifold has a rigid center at the origin.…”

“…In essence, system (1) has a rigid center at the origin when its restriction to the center manifold has a rigid center at the origin. Moreover, in [16], the authors solved the Center Problem for several families of rigid systems.…”

“…Our goal in this work is to obtain lower bounds for the quantity of small amplitude limit cycles which bifurcate from the rigid centers identified in [16]. The structure of this paper is as follows: In Section 2, we present the preliminary results on the investigation of the cyclicity for three-dimensional systems as well as an overview of the concept of rigid systems studied in [16]. In Section 3 we study the cyclicity of the rigid centers considered in [16] obtaining lower bounds for the cyclicity of each system under quadratic perturbations.…”

“…The structure of this paper is as follows: In Section 2, we present the preliminary results on the investigation of the cyclicity for three-dimensional systems as well as an overview of the concept of rigid systems studied in [16]. In Section 3 we study the cyclicity of the rigid centers considered in [16] obtaining lower bounds for the cyclicity of each system under quadratic perturbations. Also in this section, we prove the main result of this work in which we present an example of a quadratic three-dimensional system from which is possible to obtain 13 small limit cycles from quadratic perturbations.…”