Limit cycles bifurcating from isochronous surfaces of revolution in \(\mathbb R^3\)
Consider a differential system of the form \[ \dot x= X_0 (x) + \varepsilon X(x), \tag{1} \] where \(x \in \mathbb{R}^3\), \(X_0\), \(X: \mathbb{R}^3 \to \mathbb{R}^3\) are vector fields and \(\varepsilon\) is a real small parameter, and the dot denotes the derivative with respect to time. If we suppose that, for \(\varepsilon=0\), (1) has an isochronous invariant surface \(S\), that is, \(S\) is foliated by periodic orbits with the same period, then the following natural questions arise: For \(\epsilon \neq 0\) sufficiently small, does the differential system (1) posses limit cycles emerging form the periodic orbits of \(S\)? How to compute them and what about their number? A tool for treating these problems is the averaging theory. However, for a system of form (1) it is not possible to apply the classical averaging theory. In this paper, a result that goes back to \textit{I. G. Malkin} [Some problems of the theory of nonlinear oscillations (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0070.08703)] and \textit{M. Roseau} [Vibrations non linéaires et théorie de la stabilité. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0135.30603)] is applied and adapted to the problems that are studied in the paper. The authors study the number of limit cycles bifurcating from isochronous surfaces of revolution in \(\mathbb{R}^3\), when polynomial perturbations of arbitrary degree are considered.
scientific article; zbMATH DE number 5904480
scientific article