Limit cycles for nonsmooth differential equations via Schwarzian derivative
Consider autonomous two-dimensional differential systems \[ dx/dt = P(x,y), \qquad dx/dt = Q(x,y) \tag \(*\) \] where \(P\) and \(Q\) are polynomials which can have a discontinuity at the \(x\)-axis. The authors study the maximal number of limit cycles of \((*)\). Under the assumption \(P\) and \(Q\) to be homogeneous polynomials satisfying some additional hypotheses, it is proved that \((*)\) possesses at most one singular limit cycle and two regular limit cycles where the number of the multiplicities of the regular cycles is not greater than two. A key point in the proof is the study of the Schwarzian derivative of the return map.
scientific article; zbMATH DE number 1000160