On the crossing limit cycles created by a discontinuous piecewise differential system formed by three linear Hamiltonian saddles

Abstract

We study planar discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by the non-regular line Σ = { ( x , y ) ∈ R 2 : ( y = 0 ) ∨ ( x = 0 ∧ y ≥ 0 ) } . We prove that when the linear Hamiltonian saddles are homogeneous they have no limit cycles, and when they are non homogeneous they can have at most three limit cycles having exactly one point on each branch of Σ, and at most one limit cycle having four intersection points on Σ. Moreover we show that they can have at most one limit cycle having four intersection points on Σ and three limit cycles having three intersection points on Σ, simultaneously. Thus we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems. Furthermore we show that for the three types of combinations of the limit cycles here studied in two of them the upper bound is sharp by providing examples of these systems with the maximum number of possible limit cycles. For the remaining type of limit cycles the upper bound is four and we have examples with three limit cycles. So at this moment it is an open problem to know if the sharp upper bound is either four or three.