Frequency of the relative phase at interpersonal distances of 2.70–3.00 m.

<p>The frequencies of the relative phase per 0.10-m interval at interpersonal distances of 2.70–3.00 m were calculated, and the averages and standard deviations are presented (2.70–2.80 m, 2.80–2.90 m, 2.90–3.00 m). Relative phases were divided into nine ranges (20°: 0–20°, 40°: 20–40°,... 180°: 160–180°). The red asterisks above the red line indicate distances that were significantly more frequent than were those at 160° and/or 180°. The black asterisks below the lines indicate significant differences between 2.70–2.80 m and 2.90–3.00 m in each relative-phase range.</p

Frequency of relative phase at each interpersonal distance.

<p>The frequency of relative phase per 0.30 m was calculated for each match, and the averages and standard deviations are indicated. <b>A</b>. Total frequency at all distances (0.60–3.60-m). <b>B–K</b>. Partial frequencies at each distance (0.60–0.90 m, 0.90–1.20 m,... 3.30–3.60 m). Relative phases were divided into nine ranges (20°: 0–20°, 40°: 20–40°,... 180°: 160–180°). The black asterisks (*<i>p</i><.05) above the line indicate distances that were significantly more frequent than were those at 20° and/or 40°, and the asterisks below the line indicate distances that were significantly more frequent than those at 160° and/or 180°.</p

Schematic representation of step toward–away velocity (SV).

<p>In this example, player A moved from position <i>A</i>(<i>t</i>−1) to <i>A</i>(<i>t</i>+1) during <i>t</i> time from <i>t</i>−1 to <i>t</i>+1. <i>SV</i> for player A is depicted as <i>PA</i>(<i>t</i>) (red solid arrow) and was defined as the projection of vector <i>A</i>(<i>t</i>−1) → <i>A</i>(<i>t</i>+1) to vector <i>LA</i>(<i>t</i>) (red broken arrow). A negative sign of <i>SV</i> was assigned if the <i>SV</i> was directed toward the opponent, and a positive <i>SV</i> was assigned if the <i>SV</i> was directed away from the opponent. <i>SV</i> for player B was calculated using the same procedure (blue solid arrow). Note: Red and blue solid arrows denote the magnitude and direction of <i>SV</i>s for each player at <i>A</i>(<i>t</i>) and <i>B</i>(<i>t</i>), separately. In this example, the players moved primarily back and forth (along the <i>y</i>-axis) while facing each other. Although rare, lateral movements (along <i>x</i>-axis) were also observed and were included in the vector calculation.</p

Well fitted scenes switching between different functions in each scene.

<p>The numbers show the different functions in each scene.</p

Scene selection.

<p>(<b>A</b>) Kendo match. (<b>B</b>) Trajectories of two players during a kendo match over a 5-min period in a two-dimensional plane (). (<b>C</b>) Time series of interpersonal distance (IPD) for one match. (<b>D</b>) Time series of IPD for one sequence eliminating unrelated scenes. (<b>E</b>) Time series of IPD for one scene that begins with the contestants at the greatest distance from one another and ends with them coming together in a striking action.</p

Frequency of interpersonal distance.

<p>The frequency of interpersonal distance per 0.10 m in each match was calculated, and the average and standard deviation are shown (0.50–3.90 m). For example, an interpersonal distance in the 0.60–0.70-m range means that the data fell between 0.60 and 0.70 m. The asterisks (*<i>p</i><.05) above the line indicate distances that were significantly less frequent than were those within the range of 2.50–2.80 m. The red dotted line indicates the average of possible striking distances.</p

State transition diagrams.

<p>(<b>A, B</b>) Return maps were plotted using observed points as four different linear functions of an attractor (red) and a repeller (blue) for expert and intermediate competitors respectively. The circles show crossing points with the line of identity, . (C, D) Red, blue, and black lines show histograms of crossing points for an attractor, a repeller, and the sum of these respectively. (E, F) Second-order state transition diagrams with the conditional probabilities consisted of the “farthest apart” high velocity state (F) and the “nearest together” low velocity state (N) for expert and intermediate competitors, respectively. (G, H) The third-order state transition diagrams comprised four second-order sub-states.</p

Trajectories of six functions and return map analysis.

<p>(<b>a–d</b>) Linear functions, , with four different slopes for and , respectively. (<b>e</b>) Exponential function, . (<b>f</b>) Logarithmic function, . (<b>a</b>) Asymptotic trajectory to the attractive fixed point as a series of points, which corresponds to the movement of decreasing IPD by the step-towards motion shown in (<b>A</b>). (<b>a’</b>) Observed series of points in a scene from to , approaching an attractor with . (<b>b</b>) Rotational trajectory to the attractor, which corresponds to the movement of decreasing IPD by alternating step-towards and step-away motions shown in (<b>B</b>). (<b>b’</b>) Observed series of points in a scene from to , approaching an attractor with . (<b>c</b>) Diverging from the repellent fixed point asymptotically, decreasing IPD by the step-towards motions shown in (<b>C</b>). (<b>c’</b>) Series of points ( to ), diverging from repeller with . (<b>d</b>) Diverging from the repeller rotationally, increasing IPD by alternating step-towards and step-away motions shown in (<b>D</b>). (<b>d’</b>) Series of points ( to ), diverging from a repeller with . (<b>e</b>) Approaching and diverging trajectories around the attractor and/or the repeller exponentially, increasing IPD by step-away motions shown in (<b>E</b>). (<b>e’</b>) Series of points ( to ), diverging from a repeller with . (<b>f</b>) Logarithmically approaching and diverging trajectories around an attractor, decreasing IPD by step-towards from motions shown in (<b>F</b>). (<b>f’</b>) Series of points ( to ) approaching an attractor and diverging from a repeller ( to ) with .</p

Switching Dynamics in an Interpersonal Competition Brings about “Deadlock” Synchronization of Players

<div><p>In competitive sport game behavior, certain interpersonal patterns of movement coordination evolve even though each individual player only intends to exert their own strategy to win. To investigate this interpersonal pattern formation process, we asked pairs of naïve participants to engage in a play-tag game in which they had to remove a tag fastened to their partner's hip. Relative phase analysis of the players' step towards-away velocities indicated that anti-phase synchronization evolved across 10 repetitions of the game. We clarified evolution of this synchronization process using a dynamical model with an attractor (at relative phase) and a repeller (at relative phase) and discuss the self-organized nature of model and its ability to embody general solution for martial art interpersonal coordination.</p> </div

Phase lag () and a role of winner at which either player's tag was taken in the five pairs.

<p>Red triangle markers denote winner stepped toward (i.e. predator role) and blue triangle denotes winner stepped away (i.e. prey role).</p