Dulac time of a resonant saddle in the loud family
For the normalized Loud family of reversible systems \[\begin{aligned} \dot{x} &=-y+xy \\ \dot{y} &=x+Dx^2+Fy^2, \end{aligned}\] the author justifies the behavior of the period function on the outer boundary of the period annulus which surrounds the center at the origin. Namely, taking \(\mu=(D,F)\in W:=(-1,-\frac12)\times (\frac13,\frac23)\), the first few terms in the expansion of the related period function \(T(\mu,s)\) are calculated. Then using them and uniformity with respect to \(\mu\), it is proved that \(T(\mu,s)\) is regular and moreover \(\partial_sT<0\) in a neighborhood of the segment \(S:=W\cap\{F=\frac12\}\). In such a way the role of \(S\) in the bifurcation diagram (as given initially by \textit{P. Mardešić} et al. [J. Differ. Equations 224, No. 1, 120--171 (2006; Zbl 1092.34020)] is clarified.
scientific article; zbMATH DE number 7216735
scientific article