Criticality via first order development of the period constants
Consider a two-dimensional analytic real differential system \[\dot{x}=P(x,y),\quad \dot{y}=Q(x,y)\tag{\(\ast\)}\] which has a nondegenerate center-focus at the origin. One interesting problem is to study the number of oscillations of the periodic function, also known as critical periods, which is related to subharmonic bifurcations. If the origin is a center of system \((\ast)\) and if \(T(\rho)\) is the period of the periodic orbit passing through the point \((\rho,0)\), then \(T(\rho)\) is called the period function. The value \(T(\rho^{*})\) is called a critical period of the system if \(T^{\prime}(\rho^*)=0\). In this paper, the authors analyze the bifurcation of local critical periods in the class of reversible polynomial vector fields. They denoted the maximum number of local critical periods that can bifurcate from the origin of an \(n\)th degree reversible planar system, by \(\mathcal{C}_{\textit{l}}(n)\). The technique used in this paper is based on the equivalence of the first Melnikov function for the period of perturbation of an isochronous system and the linear developments with respect to the perturbation parameters of the period constant also near the same isochronous system. With this method, the authors provide the lower bound of \(\mathcal{C}_{\textit{l}}(n)\) for some even values of the degree \(n\), more preciselly they prove that the number of local critical periods in the family of polynomial time-reversible systems of degree \(n\) is at least \(\kappa(n)\), i.e., \(\kappa(n)\leq \mathcal{C}_{\textit{l}}(n)\), where \(\kappa(4)=10\), \(\kappa(6)=22\), \(\kappa(8)=37\), \(\kappa(10)=57\), \(\kappa(12)=80\), \(\kappa(14)=106\) and \(\kappa(16)=136\).
scientific article; zbMATH DE number 7312790
scientific article