Polyrhythmic multifrequency synchronization in coupled oscillators with exactly solvable attractors

Abstract

While stable polyrhythmic multifrequency [Formula: see text] dynamics has traditionally been an important element in music performance, recently, this type of dynamics has been discovered in the human brain in terms of elementary temporal neural activity patterns. In this context, the canonical-dissipative systems framework is a promising modeling approach due to its two key features to bridge the gap between classical mechanics and life sciences, on the one hand, and to provide analytical or semi-analytical solutions, on the other hand. Within this framework, a family of testbed models is constructed that exhibit [Formula: see text] multifrequency limit cycle attractors describing two components oscillating with frequencies at [Formula: see text] ratios and stable polyrhythmic phase relationships. The attractors are super-integrable due to the existence of third invariants of motion for all [Formula: see text] ratios. Strikingly, all [Formula: see text] attractors models satisfy the same generic bifurcation diagram. The study generalizes earlier work on super-integrable systems, on the one hand, and canonical-dissipative limit cycle oscillators, on the other hand. Explicit worked-out models for 1[Formula: see text]:[Formula: see text]4 and 2[Formula: see text]:[Formula: see text]3 frequency ratios are presented.