Multirhythmicity but no deterministic chaos in vibrating strings

Abstract

• We the dynamics of freely vibrating guitar strings. • We use methods of nonlinear time series analysis. • We show how false results can be mistaken as proof of deterministic chaos. • We show how to avoid false claims of chaos in vibrating strings. • We note the applicability of our approach in classrooms and for other musical instruments. The determination of properties of nonlinear dynamics from recorded time series offers fascinating insights into everyday phenomena that can be explored in the lab as well as in physics classrooms. A noteworthy challenge thereby is the distinction between multirhythmic signals, i.e., time series with many but still a finite number of harmonic frequencies, and deterministically chaotic signals. Especially if the signals are relatively short and decaying in intensity, both can have very similar continuous Fourier transforms. We show that stringed instruments offer easily accessible waveforms that can be studied with methods of nonlinear time series analysis to demonstrate that even highly multirhythmic signals can be robustly distinguished from deterministic chaos. In particular, we study the transition from sounds to chords by means of recordings of picked guitar strings, showing how the increasing complexity of the waveforms is still in fact periodic although it might seem chaotic. The treatment is instructive in terms of identifying common pitfalls that are associated with declaring deterministic chaos in observed data, and since it relies on readily available resources, it seems particularly well suited for the classroom on nonlinear dynamics and time series analysis.