This research reported here draws on self-organization theories and dynamical system models to investigate the collective behaviour of tennis players. In tennis, the unceasing to-and-fro displacements of a player about a "home" reference position, located in the middle of the baseline, are akin to those of an oscillator, and the reciprocal attending of both players establishes an informational linkage. Thus, theoretically, the displacement of the two players can be analysed as a system formed by two coupled non-linear oscillators. In such a system, relative phase has been shown to be an adept measure of the temporal synchronization between the oscillators. We hypothesized that relative phase is a relevant collective variable to characterize the relative motion of tennis players. Four players were videotaped and their displacements analysed. The results revealed just two stable patterns of synchronization, in-phase and anti-phase, as the players moved in the same or opposite directions, respectively. Moreover, relative phase showed two types of evolution within trials: either it remained stable at in-phase or anti-phase, or it exhibited transitions between these two modes. In accordance with our hypothesis, the results identified relative phase as a pertinent collective variable to represent both invariance and change in the relative displacements of tennis players. Such a finding opens new avenues for investigating dual sports.

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