Analytic obstructions to isochronicity in codimension 1 (Q2866542)

A germ of a planar analytic family of ordinary differential equations depending on one real parameter \(\eta\) of the following form is considered: NEWLINE\[NEWLINE \begin{aligned} \dot{x} & = \alpha(\eta) x - \beta(\eta) y + \sum_{j+k \geq 2} b_{jk}(\eta) x^j y^k \\ \dot{y} & = \beta(\eta) x +\alpha (\eta) y + \sum_{j+k \geq 2} c_{jk}(\eta) x^j y^k, \end{aligned} NEWLINE\]NEWLINE where \(\alpha\), \(\beta\), \(b_{jk}\) and \(c_{jk}\) are real analytic and \(\beta(0) \neq 0\). This system has a non degenerate monodromic singular point at the origin for all \(\eta\) close to \(0\). The question risen in this paper is when the origin \((x,y)=(0,0)\) of this parametrical family of systems is isochronous, that is, when there exists a cross-section through the origin which is cut at constant time intervals by the flow of the system for each fixed \(\eta\) close to \(0\). \newlineNEWLINENEWLINEIn the case that \(\alpha(0) \neq 0\), the origin is a strong focus and by the Poincaré linearization theorem, the origin is always isochronous. From now on, it is assumed that \(\alpha(0) \, = \, 0\) and the generic condition \(\alpha'(0) \neq 0\). In the case that the origin is a linearizable center for all \(\eta\) close to \(0\), one can also prove that the origin is isochronous. The case in which the origin is a weak focus is much more involved. \newlineNEWLINENEWLINEThe paper is written in five sections. The first one is devoted to the main definitions. The second section deals with the isochronicity problem explaining its state of the art. The author covers in this section mainly his contributions in [J. Differ. Equations 253, No. 6, 1692--1708 (2012; Zbl 1277.34040)]] and [\textit{W. Arriagada-Silva} and \textit{C. Rousseau}, Ann. Fac. Sci. Toulouse, Math. (6) 20, No. 3, 541--580 (2011; Zbl 1242.58021)]. The main tools used to deal with this problem are normal form surgery, monodromy of the singular point and Glutsyuk invariants, in particular the formal temporal invariant. The third section of the paper contains the main results of the paper. One of this results establishes that the origin is isochronous if and only if the temporal part is trivial and the formal temporal invariant vanishes identically. The fourth section provides several examples which are Darboux integrable. The last section contains several final remarks. This is, possibly, the first paper in which the temporal normalizability and isochronicity are related. These invariants appear in the analytic classification of parametrical families of differential systems.