Timing and Phase Locking in Cascade Juggling

Peter J. Beek*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


A natural-physical approach is pursued in uncovering basic timing and phase relations in human rhythmic movement. The approach is based on the theory of nonlinear oscillatory motion, entrained by continuously and discretely distributed forcing. In the context of juggling three balls in a figure-eight pattern, a preliminary modeling attempt of the cyclical hand motion suggested that the dynamics underwriting juggling are captured best by a discretely kicked, highly nonlinear, self-sustained oscillator. Discretely kicked, nonlinear oscillators may be characterized by regime diagrams that depict the periodic (phase-locked) and quasiperiodic (not phase-locked) regimes in which the system can operate depending on the magnitude of the kicks. This article provides evidence for 2-quasiperiodicity and near, but not perfect, phase locking between tl/tf and tu/tf (where tl is the mean time that the hands move loaded with a ball, tu is the mean time that the hands move empty, and tf is the mean flight time of the balls). Jugglers perform along the boundaries of Arnol'd tongues (representing complete phase locking) in a regime diagram without actually entering into them. With the help of Denjoy's decomposition of phase modulation into a fast and a slow mode, the deviation from the potential minimum defined by complete phase locking can be understood. The frequency ratios within the continuous relative phase between the two juggling hands reveal a Farey type of phase-locking structure, allowing a qualitative insight into which regimes jugglers position themselves when asked to speed up or slow down their act. Modulation of the hand movements increases when timing constraints become more severe (e.g., when the number of balls in the air increases). The modified standard map promises to be an adequate tool in analyzing the phase progression in juggling. All in all, the results favor an understanding of rhythmic movement in terms of discretely forced, nonlinear dynamics, rather than fully autonomous, self-sustaining oscillators.

Original languageEnglish
Pages (from-to)55-96
Number of pages42
JournalEcological Psychology
Issue number1
Publication statusPublished - 1 Mar 1989


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