Limit cycles of discontinuous piecewise smooth differential systems in the plane R2 and in the space R3

Abstract

The study of the limit cycles is one of the most important objectives in the qualitative theory of the planar ordinary differential equations. We remark that to obtain an upper bound for the maximum number of limit cycles for a given differential system in the plane R2, in general, is a very difficult problem. The study of the discontinuous piecewise differential systems, more recently also called Filippov systems, has attracted the attention of the mathematicians during these past decades due to their applications. These piecewise differential systems in the plane are formed by different differential systems defined in distinct regions separated by a curve. A pioneering work on this subject was due to Andronov, Vitt and Khaikin in 1920's, and later on Filippov in 1988 provided the theoretical bases for this kind of differential systems. As for the smooth differential systems the study of the existence and location of limit cycles in the piecewise differential systems is also of great importance. The main tools for computing analytically limit cycles of differential systems are based on the averaging theory, the Melnikov integral, the Poincaré map, the Poincaré map together with the Newton-Kantorovich Theorem or the Poincaré-Miranda theorem, and the use of the first integrals of the differential systems for computing their limit cycles. The objective of this project will be the study of the limit cycles of some relevant classes of discontinuous piecewise differential systems in the plane R2 and in the space R3. Our tools for studying such limit cycles will be the averaging theory when we do not know the first integrals of the differential systems which form the discontinuous piecewise differential systems, and the first integrals when these are known. (AU)