A numerical approximation of the rotation number
This paper is concerned with the numerical computation of the rotation number of homeomorphisms \(f : S^1 \to S^1\) on the circle. As it is well known this number allows to obtain important properties of the orbit defined by \(f\), thus it is a rational number if and only if \(f\) defines a periodic orbit and if it is irrational the orbit is dense on \(S^1\). Moreover the study of rotation numbers of a parameter dependent family of diffeomorphisms allows to explain the dynamical behaviour of bifurcations [see e.g. \textit{J. Guckenheimer} and \textit{P. Holmes}, Nonlinear oscillations, dynamical systems and bifurcations of vector fields (1983; Zbl 0515.34001)]. The author proves a theorem which states some computable conditions that allow to determine intervals of existence of the rotation number in the case that it is an irrational number and its exact value in the rational case. As it is expected, smaller intervals of existence of the rotation numbers can be obtained at the price to increase the computational cost. Some numerical experiments with maps on the circle which are known to possess rational and irrational rotation numbers are presented to show the behaviour of the results given in the theorem. Finally, the same algorithm is applied to a family of two-dimensional maps, the delayed logistic map, and it is found that also in this case the rotation numbers are computed with remarkable accuracy.
scientific article; zbMATH DE number 847554
scientific article