Rigorous enclosures of rotation numbers by interval methods

TL;DR

This paper introduces an interval-based numerical approach to accurately enclose rotation numbers of circle maps, including a method to verify the existence of periodic points, demonstrated through numerical experiments.

Contribution

It presents a novel interval method for computing and verifying rotation numbers and periodic points in circle maps, enhancing precision and rigor over previous techniques.

Findings

Interval methods produce accurate enclosures of rotation numbers.

The approach effectively verifies the existence of periodic points.

Numerical experiments demonstrate the method's practical effectiveness.

Abstract

We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.