Cyclicity of a critical point and the inverse Poincaré map
The author investigates the existence of periodic solutions to equations \((E)\) of the form \[ {d\rho\over d\theta}= \sum^\infty_{j=2} \alpha_j(\theta) \rho^j. \] This study is motivated by a relationship existing between the periodic solutions to the above equations and the cycles of plane systems of the type \(\dot x= -y+ P(x,y)\), \(\dot y= x+Q(x,y)\), the partial derivatives of \(P\) and \(Q\) vanishing at the origin. A necessary and sufficient condition for all small-amplitude solutions to \((E)\) to be periodic is given. An upper estimate to the number of small-amplitude bifurcating periodic solutions to \((E)\) is given. Such results are applied to study periodic solutions to systems with constant angular speed, \(\dot x= -y+ x(ax+ by+ p(x,y))\), \(\dot y= x+ y(ax+ by+ p(x,y))\), \(p(x,y)\) being a homogeneous polynomial. Necessary and sufficient conditions for such systems to have a center are given (since the angular speed is constant, such centers are isochronous; this is not observed in the paper). It is also proved that, if the degree of \(p(x,y)\) is \(k\), then at most \([k/2]\) periodic solutions can bifurcate from the origin.
scientific article; zbMATH DE number 1539289