Scientific Collaboration Between the UIB Dynamic Systems Group and the UNESP Dynamic Systems Group

The Dynamic Systems Group at the University of the Balearic Islands (UIB) has initiated a scientific collaboration with the Dynamic Systems Group at São Paulo State University Julio de Mesquita Filho (UNESP) at Sao José de Rio Preto. This collaboration focuses on the qualitative theory of differential equations and aligns with the objectives of the PID2020-118726GB-I00 project, overseen by Rafel J. Prohens and José Luis Bravo, and the “Piecewise Smooth Vector Fields” project, overseen by Professors Paulo Ricardo da Silva and Claudio Aguinaldo Buzzi.

The collaboration has thus far included participation in the XII Workshop on Dynamical Systems, held from November 6-9, 2023, at the IBILCE center at UNESP, by Professors M. Jesús Álvarez, Tomeu Coll, and Rafel J. Prohens. Additionally, Professor Antonio E. Teruel conducted a two-week scientific visit to the IBILCE center at UNESP.

During these activities, the SSDD-UIB group presented the following works:

Title: Rigid systems in the plane. Overview and new results.
Date: November 6, 2023
Speaker: M. Jesús Álvarez
Abstract: In this presentation, we provide an overview of the current state of knowledge concerning rigid systems in the plane. Additionally, we present new results for a family of this class of systems.
Rigid system are characterized by having the origin as its unique critical point (which is always monodromic) and by all orbits around it having constant angular velocity. There are several factors that make the rigid family interesting. Firstly, the fact that the origin is its only critical point implies that any potential limit cycles, if they exist, have to be nested around it. Secondly, this family plays an important role in the broader problem of isochronicity.
Our presentation focuses on the quartic rigid family, that is the simplest non-trivial polynomial family of rigid systems without rotatory parameters. We study this family within the finite plane and on the Poincaré sphere. Within this family we prove that for a significant subfamily there are no limit cycles in the plane. However, when the critical points at infinity are cusps (the generic case), a periodic orbit crossing the equator of the sphere always exists, when no homoclinic nor heteroclinic orbits exist.

Title: Probabilty of existence of limit cycles for a family of planar systems.
Date: November 9, 2023
Speaker: Rafel J. Prohens
Abstract: The goal of this work is the study of the probability of occurrence of limit cycles for a family of planar differential systems that are a natural extension of linear ones. To prove our results we first develop several results of non-existence, existence, uniqueness and non-uniqueness of limit cycles for this family. They are obtained by studying some Abelian integrals, via degenerate Andronov-Hopf bifurcations or by using the Bendixson-Dulac criterion. To the best of our knowledge, this is the first time that the probability of existence of limit cycles for a non-trivial family of planar systems is obtained analytically. In particular, we give vector fields for which the probability of having limit cycles is positive, but as small as desired.

Title: Saddle Node Canard Cycles in PWL Systems,
Date: May 22, 2024
Speaker: Antonio E. Teruel
Abstract: This work delves into and enhances existing knowledge on canard explosions in planar slow-fast systems using piecewise linear systems. The presented results cover both supercritical and subcritical scenarios, with an emphasis on the existence of non-hyperbolic canard orbits of the saddle-node type.

Title: Slow Passage through a PWL Transcritical Bifurcation,
Date: May 27, 2024
Speaker: Antonio E. Teruel
Abstract: This work fully describes the behavior of the maximal delay as a function of parameters in the context of slow passage through a transcritical bifurcation defined by a piecewise linear system. The obtained results are used to analyze the behavior of a piecewise linear system that exhibits two transcritical bifurcations, leading to an unbounded canard explosion, which culminates in an enhanced delay phenomenon.