Multifrequency behavioral patterns and the phase attractive circle map

Abstract

With relative phase as a collective variable or order parameter, phase attractive dynamics can capture the temporally coherent behavior of a large number of different experimental systems. We present results from multifrequency coordination experiments in humans showing: a) that phase attraction persists especially for low order frequency ratios; b) that short-term jumps from one phase relation to another occur within a frequency ratio; c) that the most stable frequency-ratios are low order; and d) that transitions frequently occur from higher order (e.g. 5:2, 4:3) to lower order (2:1, 1:1) frequency ratios. We study a modified sine circle map with built-in phase attractive dynamics that qualitatively accounts for these results. In this phase-attractive map, patterns arise from competition between external driving and intrinsic phase attractive dynamics. The relative strength of extrinsic and intrinsic parameters determines the width of Arnol'd tongues, thereby influencing the delay or acceleration of irregular behavior. Behavioral complexity is inversely proportional to tongue width, thus accounting for the relative difficulty of performing different multifrequency behaviors and why "errors" in such behavior are often seen to occur.