Perturbed rigidly isochronous centers and their critical periods (Q2396874)

Consider a planar system which has a center at the origin. Call \(T(\eta)\) the period of the orbit passing through \((\eta,0)\). When \(T'(\eta^*)=0\), for some \(\eta^*\), the period \(T(\eta^*)=0\) is called \textit{critical}. Also recall that a center is \textit{isochronous} if the function \(T\) is constant and \textit{rigidly isochronous} if in polar coordinates \(\dot \theta\equiv 1\) (\(\theta\) being the angular coordinate). This paper is concerned with the bifurcation of critical periods from the orbits of a rigidly isochronous center under small polynomial perturbations of degree \(n\). Upper estimates are found for the number of arising critical periods for values of \(n\geq 3\). Such estimates are proven to be sharp (in the sense that they can be reached) for some values of \(n\).