(untitled) (Journal of Nonlinear Modeling and Analysis, 2024)
During this century, an increasing interest appeared in studying the planar piecewise differential systems. This is due to their numerous applications for modelling many natural phenomena. For understanding the dynamics of the planar differential systems we must control the existence or non-existence of periodic orbits and limit cycles. So many papers have been published studying the existence or non-existence of periodic orbits and limit cycles for continuous or discontinuous piecewise differential systems. But until now very few papers have studied the periodic orbits and limit cycles of piecewise differential systems where two differential systems of the piecewise differential system are continuous and discontinuous respectively. We study the periodic orbits and limit cycles of the planar continuous–discontinuous piecewise differential systems separated by two parallel straight lines, such that either in one of these straight lines the piecewise differential system is continuous and in the other one discontinuous. In two pieces of these piecewise differential systems there are arbitrary Hamiltonian systems of degree two and in the third piece there is an arbitrary Hamiltonian system of degree one forming the continuous-discontinuous piecewise differential systems. We determine the limit cycles of these piecewise differential systems by considering two cases. In the first the Hamiltonian system of degree one can be in the middle of the three zones, and in the second it is on one side of the three zones.