Bifurcation of local critical periods in the generalized Loud's system (Q428085)

The problem of isochronous center is solved for the system NEWLINE\[NEWLINE\dot x= -y+ Bx^{n-1} y,\qquad\dot y= x+ Dx^n+ Fx^{n-2} y^2,NEWLINE\]NEWLINE where \(B,D,F\in\mathbb{R}\) and \(n\geq 3\) is a fixed natural number. For \(n\) even, it is shown that at most two critical periods bifurcate from a weak center of finite order or from the linear isochrone, and at most one local critical period from \(q\) nonlinear isochrone. For \(n\) odd, it is proved that at most one critical period bifurcates from the weak centers of finite or infinite order. It is also shown that the upper bound is sharp in all the cases. (For \(n = 2\), this was proved by \textit{C. Chicone} and \textit{M. Jacobs} [Trans. Am. Math. Soc. 312, No. 2, 433--486 (1989; Zbl 0678.58027)] and the author's proof strongly relies on their general results about the issue.)