Limit cycles of piecewise polynomial differential systems with the discontinuity line \(xy = 0\)
The authors study the family of perturbed planar systems \[ \dot x= -yF(x,y)+\varepsilon P(x,y,\varepsilon),\quad \dot y=xF(x,y)+\varepsilon Q(x,y,\varepsilon) \] where \(F=H^m, m\ge0,\) \(H(x,y)=(x^2+y^2)/2\) and \(P\) and \(Q\) are piecewise polynomial perturbations, with discontinuities at \(xy=0.\) Notice that the unperturbed system has the first integral \(H\) and a global center at the origin. This fact makes it very suitable to be studied by using the averaging theory. The paper gives upper bounds for the number of limit cycles appearing by using the first non identically null averaged function in terms of the degree of the perturbation and the order of this averaged function. These bounds are sharpened either by assuming that the perturbations are hamiltonian or that these perturbations have some special symmetries.
piecewise polynomial differential equations
scientific article; zbMATH DE number 7436400
scientific article