Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

The problem of determining the existence, maximum number and positions of the limit cycles of the planar discontinuous piecewise linear differential systems is an important problem in the qualitative theory of differential systems. In this paper, we study two families of piecewise linear Hamiltonian systems without equilibria in R-2 separated by a nonregular curve. We provide the maximum number of crossing limit cycles that each family can have and show when this maximum is reached. In this way we are solving for each family the extended 16th Hilbert problem.