Figure 1.
The experimental task: a 3 vs. 1 ball possession task in soccer practice for studying interactions among three people.
Three attackers (Attackers 1, 2, 3) were asked to pass a ball as much as possible, without allowing it to be intercepted by one defender for 90 s in a 6-m square. Angular oscillations were constructed by the three attackers (,
,
).
Figure 2.
Five predicted patterns of a ring of three coupled oscillators from symmetric Hopf bifurcation theory.
The number in the circle indicates the number of the waveforms. The double circle shows that the corresponding oscillator has double frequency. Relationships between two oscillators are indicated by = : in-phase, phase shift,
anti-phase.
Figure 3.
Three synchronized patterns in a 3 vs. 1 ball possession task as rings of three coupled oscillators predicted by symmetric Hopf bifurcation theory.
(A, C, F) Time series of angles. (B, D, G, E, H) Trajectories on the phase plane. (B), (D), and (G) show the phase plane of time series in (A), (C), and (F), respectively. (+) in the phase plane shows that all three oscillators would have equally. (A, B) A rotation pattern (R) in which all three oscillators are synchronized while keeping the phase difference
. (C-E) A partial anti-phase pattern (PA) in which two oscillators are synchronized in anti-phase and another is constant. PA1 shows the case in which the constant value is smaller than
and the value of PA2 is larger than that of
. (F-H) A partial in-phase pattern (PI) in which two oscillators are synchronized in-phase and another is in anti-phase synchronization.
and
show the cases in which one of the two in-phase oscillators is larger than
and the other one is smaller than
and vise versa.
Figure 4.
Comparison between the experimental data and predicted patterns.
(A-C) Color contour plots show experimental time-frequency trajectory plots on the phase plane. (A), (B), and (C) show the high-, mid-, and low-level groups, respectively. The color indicates the height normalized by the maximum and minimum frequency values for each skill level. Dark red is the highest and dark blue is the lowest, while white represents no trajectories. (D, E) Two predicted trajectories on the phase plane selected from Figure 2. (D) shows a rotation pattern (R) that is similar to (A) for the distribution of the high-level group. (E) shows a partial anti-phase pattern () that is similar to (B) and (C) for the mid and low levels, respectively.
Figure 5.
Typical examples of the high- and low-skill-level over 10 s.
(A, B) Time series of three angle oscillations. The scale of angle A was reduced so that the scale in (A) could be three times smaller than that in (B). (*) in the time series indicates peaks. (C, B) Phase difference between any two angles. These values were calculated from the peaks in (A) and (B). (E, F) Trajectories on the phase plot for the above time series.