Integrability and piecewise planar diUerential systems

Abstract

In general, a dynamical system is everything changes over time, in mathematics is the formalization of the general scientiVc concept of a deterministic process. The future and past states of many physical, chemical, biological, ecological, economical, and even social systems can be predicted to a certain extent by knowing their present state and the laws governing their evolution. On the condition that these laws do not change in time, the behavior of such a system could be considered as completely deVned by its initial condition. Thus, the notion of a dynamical system includes a set of its possible situations (state space) and a law of the evolution of the state in time. Modelling dynamical systems by ordinary diUerential equations has a long history, see for instance [1]. Over the last hundred years, many techniques have been developed for the solution of ordinary diUerential equations but most of the non linear diUerential equations cannot be solved by the calculus methods we know at present. The qualitative theory of diUerential equations is being used to examine diUerential equations whose explicit solutions cannot be determined.These tools are originated by Henri Poincaré in his work on diUerential equations at the end of the nineteenth century [30, 31]. In this thesis, we use the qualitative theory of diUerential equations to study a diUerential system in two-dimensional and we treat the most important solution of diUerential equations which is the limit cycle introduced by H. Poincaré [31] and reported by his work the most sought solutions in the modeling of physical systems in the plane. Our Vrst aim in this thesis is to study the integrability of ordinary diUerential equations or simply diUerential systems in two real variables x˙ = P (x, y), y˙ = Q(x, y), where P and Q are polynomials of degree two. The second aim is to determine the number of limit cycles of the piecewise diUerential system of the form    x˙ = f1(x, y), y˙ = f2(x, y), H(x, y) > 0, and    x˙ = g1(x, y), y˙ = g2(x, y), H(x, y) < 0, 4 List of Figures separated by Σ = {(x, y) ∈ R 2/H(x, y) = 0}. Now, we describe the structure of this thesis which is divided into three chapters, in the Vrst one we present the necessary deVnitions, lemmas and theorems used in our study as Vxed points and their nature, Hartman-Grobman theorem, Poincaré map, piecewise diUerential system, invariant, Vrst integral..., (see [22, 23]). In chapter 2, we start to present our work by classifying quadratic diUerential systems having a special invariant of the form ax2+bxy+cy2+dx+ey+c1t, we prove that there are 21 diUerent families of quadratic systems having invariants of this form. As far as we know this is the Vrst time that quadratic diUerential systems having an invariant diUerent from a Darboux invariant have been classiVed. In the second part of this chapter, we study the limit cycles of piecewise planar diUerential systems formed by quadratic systems that have the Vrst integral of the form ax2 + bxy + cy2 and linear center. We prove that piecewise systems have a continuum of periodic orbits, so no limit cycles. In ch