Classical analytic and probabilistic methods used to predict the players` behavior in racket sports proved to yield but a contrasted and mitigated outcome. Recently, some authors proposed another approach in terms of dynamical self-organizing systems (McGarry et al., 2002, Journal of Sport Science, 20, 771-781). In this view, the game is no longer a mere addition of each player`s individual behavior, but rather a complex system issued from the interaction between the two players and the environment. Accordingly, the study of such a system requires the definition of a relevant collective variable capturing invariance and change in the coupled activity between the two players involved in the game (Haken, 1983, Synergetics, Berlin : Springer). In racket sports, the position of the two players relative to each other is of critical importance. As a first approximation, each player basically exhibits to-and-fro motion about a reference position located in the middle of the base line. Thus, we posit that relative phase (or phase lag) between the two players` displacement is a pertinent variable to characterize the various modes of collective behavior exhibited in a racket sports such as tennis. Four tennis men ranking at a national level were videotaped while they were instructed to realize long games. Forty trials lasting more than seven rallies were thus collected. After digitization, 2D displacements were decomposed according to their Cartesian coordinates. As x-motion (viz. back and forth, from the base line) was very seldom and/or of very small amplitude, only y-motion (viz. laterally, along the base line) was analyzed. A cross-correlation within a moving 5 s window was carried out between the y-motion of both players. The lag value close to a lag 0 with the most significant correlation divided the window length yielded an index of the relative phase between the two time series. Results showed that among all relative phase modes exhibited across all trials, 0° and 180° are most frequent and stable. For the first two time windows, however, relative motion hovered about 180°. Then, two evolutions of relative phase were observed. For 40% of the trials, relative phase did not change from 180° (see Fig. 1). An ANOVA with repeated measures failed to reveal any significant effect on relative phase across time windows (p > .05). For 40 other percents of the trials, relative phase exhibited a marked shift (see Fig. 2). An ANOVA with repeated measures detected a significant effect across time windows (F(5, 5) = 20.03, p < 0.01). In the last 20% of the trials, no significant trend could be detected, as relative phase never stabilized
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