Summer School on Qualitative Theory of Piecewise Ordinary Differential Equations

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.

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 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.

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).

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.