Topological imprint for periodic orbits

Abstract

The more self-crossing points an orbit has the more complex it is. We introduce the topological imprint to characterize crossing points and focus on the period-doubling cascade. The period-doubling cascade topological imprint determines the topological imprint for orbits in chaotic bands. In addition, there is a closer link between this concept and the braids studied by Lettelier et al (2000 J. Phys. A: Math. Gen. 33 1809–25).