Darboux linearization and isochronous centers with a rational first integral
The authors consider the following planar system \[ \dot x = -y + \sum_{i+j=2}^{n} a_{ij}x^i y^j ,\quad \dot y = x + \sum_{i+j=2}^{n} b_{ij}x^i y^j ,\quad\quad (x,y)\in \mathbb{R}^2 \tag{1} \] where \(a_{ij}\) and \(b_{ij}\) are real numbers. Let the system (1) has a center, and let \(T\) be a period-function, i.e. \(T(s)\) is the time taken for the trajectory starting at point \((x(s),y(s))\) to return in the same point \((x(s),y(s))\). When this period function is a constant, the center is said to be isochronous. It is known that a center is isochronous if and only if it is linearizable. The first application of isochronous oscillations is Huygen's pendulum. The authors introduced the Darboux linearization of a center and give an effective criterion for verifying Darboux linearization. Most of the known polynomial isochronous centers are Darboux linearizable. A very interesting part of the article is the determination of all Hamiltonian cubic systems which are Darboux linearizable. In the second part of the paper, the authors restrict to study the isochronous centers with rational first integrals. Here, they prove that, under certain conditions, one of the following two statements is valid: 1. The cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber, or 2. the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the center.
scientific article; zbMATH DE number 1000145
scientific article