Resumen de Four Limit Cycles of Discontinuous Piecewise Differential Systems with Nilpotent Saddles Separated by a Straight Line

Imane Benabdallah, Rebiha Benterki

• The study of discontinuous piecewise differential systems has attracted increasing attention in recent decades as a result of their applications to a variety of physical phenomena. The role of limit cycles in the study of any planar differential system is well known, but finding the maximum number of limit cycles that a class of planar differential systems can have is one of the most difficult tasks in the qualitative theory of planar differential systems. Thus, in this work, we solve the extinction of Hilbert problem for all classes of discontinuous piecewise differential systems with Hamiltonian nilpotent saddles, separated by the straight line x = 0. Firstly, we study the discontinuous piecewise differential system formed by linear center and one of the six Hamiltonian nilpotent saddles, we provide that these systems can have at most one limit cycle. Secondly, we consider all possible discontinuous piecewise differential systems formed by Hamiltonian nilpotent saddles in each piece, where we show that these 21 classes of piecewise differential systems can have at most four limit cycles.

  We have strengthened our result by giving examples for each case.