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Beek and van Santvoord [Beek, P. J., & van Santvoord, A. A. M. (1992). Journal of Motor Behavior, 24, 85-94] proposed a three-stage model of learning to juggle based on group analyses of temporal measures. Here, we examined in detail how the temporal and spatial features of juggling evolved in eight individual participants progressing from the second to the third stage of learning. During the second stage, the dwell ratio, defined as the ratio of the time that the juggler holds a ball between catch and toss and the hand cycle time (HCT), was stable when it was about 0.83. The subjects with a dwell ratio near this value and controlled throws exhibited stable juggling, whereas the subjects with a dwell ratio of 0.80 or smaller exhibited unstable juggling. Compared to the former group, the latter group had a longer time from the throw of a ball to the arrival at its zenith (TZ), and a shorter time between the arrival of an airborne ball at its zenith and the subsequent throw (IZR). The latter group also exhibited larger variability in the dwell ratio and IZR. With practice, the subjects appropriated, on average, the duration of TZ and IZR to the dwell ratio and improved the ability to accurately throw balls by changing the motions of the limb segments involved. Although these changes helped to stabilize the performance during the second stage, the variability problem was not sufficiently resolved. Only two out of eight subjects passed on to the third stage by the last (10th) Session. They achieved small variability in IZR, dwell ratio, and flight paths of the ball while juggling with short HCTs and small dwell ratios. These results suggest that the reduction of variability in these variables was essential to pass on to the third stage.[authors summary]

The relationship between point-of-gaze and ball movements in three-ball juggling was examined as a function of expertise, pattern and tempo. Five intermediately skilled and five expert jugglers performed the standard and reverse cascade at three tempos, while point-of-gaze and ball movements were recorded simultaneously. Scaled to the size of the ball patterns, the experts made smaller point-of-gaze movements than the intermediates, especially in the horizontal direction and in the standard cascade. In both skill groups, point-of-gaze and ball movements were often 1 : 1 frequency locked in the horizontal direction, whereas in the vertical direction 1 : 2 frequency locking also occurred. In the latter direction, the 1 : 1 ratio prevailed in the intermediates and the 1 : 2 ratio in the experts. In addition, the incidence of the 1 : 1 ratio decreased and that of the 1 : 2 ratio increased with increasing tempo. Furthermore, in the vertical direction, increasing tempo resulted in a weaker 1 : 1 locking, whereas the strength of the 1 : 2 ratio remained unaffected by tempo. In the horizontal direction, the strength of the 1 : 1 locking was higher on average in the reverse cascade than in the standard cascade. We conclude that expertise in juggling is reflected by an overall reduction in the extent to which the balls are visually tracked, and that task constraints such as tempo and juggling pattern affect the visual search patterns of both expert and intermediate jugglers. [authors summary]

To uncover the underlying control structure of three-ball cascade juggling, we studied its spatiotemporal properties in detail. Juggling patterns, performed at fast and preferred speeds, were recorded in the frontal plane and subsequently analyzed using principal component analysis and serial correlation techniques. As was expected on theoretical grounds, the principal component analysis revealed that maximally four instead of the original six dimensions (3 balls x 2 planar coordinates) are sufficient for describing the juggling dynamics. Juggling speed was shown to affect the number of dimensions (four for the fast condition, two for the preferred condition) as well as the smoothness of the time evolution of the eigenvectors of the principal component analysis, particularly around the catches. Contrary to the throws and the zeniths, and regardless of juggling speed, consecutive catches of the same hand showed a markedly negative lag-one serial correlation, suggesting that the catches are timed so as to preserve the temporal integrity of the juggling act. [authors summary]

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 distrib- uted 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 he 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.[author summary]

Mathematics has the power to describe, predict and create patterns, a power that is very well demonstrated in the pattern-
rich world of juggling. In this paper we examine a simple method of describing juggling patterns using mathematical
notation, and then use this notation to predict new juggling patterns. We conclude with a demonstration of how mathematics
has been used to create beautiful patterns that did not exist before these mathematical methods had been used, and how
mathematical names are now used by jugglers worldwide – a powerful demonstration of mathematics advising the arts. [author summary]

The paper studies the properties of a sinusoidally vibrating wedge billiard as a model for 2D impact juggling. It is shown that some periodic orbits that are unstable in the elastic fixed wedge become exponentially stable in the (non-)elastic vibrating wedge. These orbits are linked with some classical juggling patterns, providing an interesting benchmark for the study of the frequency-locking properties in human rhythmic tasks. [authors summary]