Example One Qualitative Theory of
Piecewise Ordinary
Differential Equations
Summer School on

Summer School on Qualitative Theory of Piecewise Ordinary Differential Equations

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Advanced course / School
From July 12, 2021
to July 16, 2021

Online via Zoom

Registration deadline 04 / 07 / 2021


♦ 30 €

* Registration is free for all affiliated researchers at the Centre de Recerca Matemàtica (CRM). Please click on the Reservation option before finishing the process.



The qualitative theory of differential equations studies the behavior of differential equations by means other than the search for their solutions. It started with the works of Henri Poincaré and Aleksandr Lyapunov. Relatively few differential equations can be solved explicitly, but using analysis and topology tools, they can be “solved” in a qualitative sense, obtaining information about their properties. Among others, the main interests in this field are the study of structural stability, bifurcations, integrability, the existence and number of periodic orbits, homoclinic and heteroclinic connections. This school will be structured in five mini-courses of 6 hours each in which the classic techniques for the study of some of the aforementioned concepts will be presented. In each course we will show the similarities and differences between smooth and non-smooth differential systems.

The summer school is supported by the European RISE project Dynamics (H2020-MSCA-RISE-2017 – 777911, http://www.gsd.uab.cat/dynamicsh2020/) and Grup de Sistemes Dinàmics de la UAB (http://www.gsd.uab.cat)

organizing committee

Jaume Llibre Universitat Autònoma de Barcelona
Gheorghe Tigan Politehnica University of Timisoara
Joan Torregrosa Universitat Autònoma de Barcelona

titles of the courses

Structural stability and bifurcations of low codimension in piecewise differential systems

by Tiago de Carvalho (Universidade de Sao Paulo)


The qualitative theory of ODE is mainly based in the classification the systems of differential equations modulo a relation of equivalence. We say that two systems are equivalent if there is a homeomorphism that sends the trajectories from one system to the other. The concept of structural stability says that a system is structurally stable if there is a neighbourhood of it such that all systems in this neighbourhood are equivalent. The set of non-structurally stable systems is called the bifurcation set.

We will study the structural stability and bifurcations in piecewise smooth systems and whenever possible compare with the smooth case. We will study the singularities of low codimension. The singularities of structurally stable systems are the codimension zero singularities. The generic singularities of non-structurally stable systems will be the codimension one singularities. The study will be essentially in dimension two, but some cases in dimension three will be presented.

Integrability and limit cycles in piecewise differential systems

by Jaume Llibre (Universitat Autònoma de Barcelona)


Nonlinear ordinary differential equations appear in many branches of applied mathematics, physics, and, in general, in applied sciences. For a differential system or a vector field defined on the plane the existence of a first integral determines completely its phase portrait, and in higher dimensions allow to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence to know first integrals is important, but a natural question arises: Given a vector field how to recognize if this vector field has a first integral?

The objective of this mini-course is double. First, we shall study how to compute first integrals for polynomial vector fields using the so-called Darboux theory of integrability. And second, we shall show how to use the existence of first integrals for computing limit cycles in piecewise differential systems.

The averaging method

by Douglas D. Novaes (Universidade Estadual de Campinas)


The averaging method is an important and celebrated method for dealing with nonlinear oscillating systems in the presence of small perturbations. It is mainly concerned with providing long-time asymptotic estimates for solutions of perturbed non-autonomous differential equations. This method has also been extensively employed in the study of periodic solutions.

We start by discussing the classical averaging method for smooth systems and its relation with a Melnikov-like procedure. Then, we shall explore results for obtaining necessary conditions for the existence of periodic solutions for smooth systems. The generalizations of such results to the nonsmooth continuous and discontinuous contexts will be fairly discussed. Finally, if there is time, we may also approach further topics, such as bifurcations from non-degenerate families of periodic solutions and torus bifurcation.

Bifurcation Analysis in Piecewise Linear Systems

by Enrique Ponce (Universidad de Sevilla)


Piecewise linear systems (PWL systems, for short) become a natural entry point in the analysis of the nonlinear dynamics to be found in more general piecewise smooth differential systems. They exhibit a lot of relevant issues both from the theoretical point of view and real applications, turning out to be accurate formulations of real engineering devices, and reasonable models for problems in different branches of bio-sciences.

We will adopt mainly the framework of bifurcation theory with especial emphasis in the mechanisms to generate limit cycles in PWL systems. To avoid long taxonomies, we will stress the importance of using adequate canonical forms in the analysis. Thus, we will review the more relevant families of PWL systems in a low-dimensional context (2D/3D), putting the emphasis on their possible bifurcations and illustrating their usefulness in the analysis of realistic applications. Apart from the analogues to ‘classical’ bifurcations in smooth differentiable dynamics, we will also revise some specific non-smooth bifurcations (for instance, boundary equilibrium bifurcations).

Center-focus and cyclicity problems

by Joan Torregrosa (Universitat Autònoma de Barcelona)


We will study the center-focus and cyclicity problems for some differential systems and piecewise differential systems in the plane. We study the stability in a neighborhood of the origin when this point is monodromic and nondegenerate.

The center-focus problem consists of how we can distinguish if the equilibrium point is a center of focus in smooth and non-smooth scenarios. We will present some of the usual techniques to study this problem in both scenarios. With this algorithm, we will show how the centers of some families can be found, and how we can solve the center problem.

The study of the number of limit cycles of small amplitude that bifurcate from an equilibrium point is known as the cyclicity problem. We will explore the known results on this local problem near weak-foci and centers families in the context of polynomial vector fields. Providing the known best lower bounds for the local smooth and non-smooth Hilbert numbers for low-degree polynomial vector fields.


A limited number of registration grants will be offered to the first members of the project who register and apply for it. Other cases will be considered on an exceptional basis.
To apply, you must complete the registration process, please go to SIGN IN, indicate in the OTHERS section that you wish to apply for a registration grant, you will be asked to attach your CV. Please click on the Reservation option before finishing the process.


Application deadline for grants is June 13, 2021
Resolutions will be sent by June 20, 2021


list of participants

Click to open
Begoña Alarcón Cotillas Universidade Federal Fluminense
Carlos Arturo Peña Rincón Universidad Sergio Arboleda
Tiago de Carvalho Universidade de Sao Paulo
Jaume Llibre Universitat Autònom de Barcelona
Alejandro Marqués Lobeiras Universidad de Oviedo
Douglas Novaes UNICAMP
Stefano Pedarra Centre de Recerca Matemàtiaca
Enrique Ponce Universidad de Sevilla
Gheorghe Tigan Politehnica University of Timisoara
Joan Torregrosa Universitat Autònoma de Barcelona
*Updated June 14, 2021


For inquiries about this event please contact the research programs coordinator Ms. Núria Hernández at nhernandez@crm.cat​​