Torsion-Adding and Asymptotic Winding Number for Periodic Window Sequences

TL;DR

This paper investigates the torsion properties of periodic windows in nonlinear dynamical systems, introduces the concept of torsion-adding, and derives a rule for the asymptotic winding number in window sequences.

Contribution

It introduces the concept of torsion-adding and provides a general rule for calculating the asymptotic winding number in periodic window sequences.

Findings

Torsion associated with periodic states varies by a constant between successive windows.

Regions of uniform torsion are identified within window sequences.

A general rule for the asymptotic winding number is derived from torsion and period-adding rules.

Abstract

In parameter space of nonlinear dynamical systems, windows of periodic states are aligned following routes of period-adding configuring periodic window sequences. In state space of driven nonlinear oscillators, we determine the torsion associated with the periodic states and identify regions of uniform torsion in the window sequences. Moreover, we find that the measured of torsion differs by a constant between successive windows in periodic window sequences. We call this phenomenon as torsion-adding. Finally, combining the torsion and the period adding rules, we deduce a general rule to obtain the asymptotic winding number in the accumulation limit of such periodic window sequences.