Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder (Q293529)

The authors investigate the persistence of limit cycles under small, discontinuous perturbations of a vector field. They assume, that the unperturbed, three-dimensional vector field has a continuous family of (unstable) synchronous limit cycles around the vertical axis, filling up a cylinder. The perturbation may have jumps along a manifold intersecting the cylinder.NEWLINENEWLINEBy applying a regularization close to the discontinuities and averaging the smoothed perturbation along the periodic solutions of the unperturbed problem, the authors construct a so-called Malkin bifurcation function \(M(z)\). If \(M(z_0)=0\) and \(M'(z_0)\neq 0\), an isolated limit cycle bifurcates at \(z=z_0\) from the invariant cylinder.NEWLINENEWLINEBy assuming, that both segments of the vector field's perturbation can be expressed as piecewise polynomials, the authors also state an upper bound for the possible number of bifurcating limit cycles.