Limit Cycles for a Class of Continuous-Discontinuous Piecewise Differential Systems (Journal of Nonlinear Modeling and Analysis, 2024)
During this century, an increasing interest appeared in studyingthe planar piecewise differential systems. This is due to their numerous applications for modelling many natural phenomena. For understanding thedynamics of the planar differential systems we must control the existence ornon-existence of periodic orbits and limit cycles. So many papers have beenpublished studying the existence or non-existence of periodic orbits and limitcycles for continuous or discontinuous piecewise differential systems. But until now very few papers have studied the periodic orbits and limit cycles ofpiecewise differential systems where two differential systems of the piecewisedifferential system are continuous and discontinuous respectively. We studythe periodic orbits and limit cycles of the planar continuous–discontinuouspiecewise differential systems separated by two parallel straight lines, suchthat either in one of these straight lines the piecewise differential system iscontinuous and in the other one discontinuous. In two pieces of these piecewise differential systems there are arbitrary Hamiltonian systems of degreetwo and in the third piece there is an arbitrary Hamiltonian system of degreeone forming the continuous-discontinuous piecewise differential systems. Wedetermine the limit cycles of these piecewise differential systems by considering two cases. In the first the Hamiltonian system of degree one can be inthe middle of the three zones, and in the second it is on one side of the threezones.
Keywords: Limit cycles; Hamiltonian system; continuous-discontinuous piecewise linear differential systems; first integrals.