## 3 Continuous to Discrete Dynamics: Return Map
### Identification of Coordination Patterns by the Return Map during Interpersonal Competition  
The first return (Lorenz) map was applied to the offensive and defensive maneuvers that occur during kendo matches as interpersonal competition [37]. Figure 3a shows the time series of state variables comprising interpersonal distance and its velocity. We detected the peaks of the time series for interpersonal distance, and the peaks of the state variables were plotted as a return map as the present peak \(X_{n}\) vs. the next peak \(X_{n+1}\) (Figure 3b).  
The periodicities on such a plot are the intersections with the identity line \(X_{n}=X_{n+1}\). These intersections are known as attractive fixed points and repellers or saddle points. These attractive fixed points are deterministically approached from a direction called the stable direction or manifold, and the repellers diverge from these attractive fixed points along the unstable direction or manifold as a linear function. Theoretically, we postulated the linear function, \(X_{n+1}=aX_{n}+b\). The intersections can be classified into two properties depending on the absolute value of \(a\). When the absolute value of \(a\) is less than 1, \(|a|\) is less than 1. Then the intersection is considered to be an attractive fixed point (i.e., an \"attractor\"). When the absolute value of \(a\) exceeds 1, \(|a|\) exceeds 1. Then the intersection is referred to as a repellent fixed point (i.e., a \"repeller\"). An attractor can be further classified into two types.  
Figure 3: Procedure for depicting the return map from the time series of state variables. (**a**) Gray, broken, and black lines show the time series for normalized \(X_{IPD}(t)\), normalized \(V_{IPD}(t)\), and \(X(t)\), respectively, for a 12-s trial with more than five peaks. The red and gray circles indicate the corresponding values of \(X(t)\) for the peaks of \(X_{IPD}(t)\). (**b**) Return map of the time series for the observed data, \(X_{n}\) vs. \(X_{n+1}\) using the amplitude of \(X(t)\) at the peaks of \(X_{IPD}(t)\) corresponding to the series of points (red and gray circles) in the panel shown in **a**. Modified from [37].  
Figure 2: Lorenz attractor and Lorenz map. Parameters \(\sigma=10,b=8/3,r=28\), and \(x_{0}=y_{0}=z_{0}=10\). (**a**) Lorenz attractor in three-dimensional space. (**b**) Time series of \(z(t)\) and peaks of \(z(t)\) are shown as red dots. (**c**) Lorenz map plotted as \(z_{n}\) vs. \(z_{n+1}\).  
When \(0<a<1\), the trajectories are asymptotically close to the attractor (Figure 4a). When \(-1<a<0\), the trajectories are rotationally close to the attractor (Figure 4b). A repeller also has two types of trajectories: \(1<a\), and \(a<-1\), corresponding to asymptotical and rotational trajectories, respectively, as shown in Figure 4c,d. Trajectories approach and diverge from points that do not cross the line \(X_{n}=X_{n+1}\). We postulated that this exponential function, \(X_{n+1}=b\ exp(aX_{n})\), and this logarithmic function, \(X_{n+1}=a\ logX_{n}+b\) (Figure 4e), represent intermittency.  
As shown in Figure 4, all six predicted types of functions were confirmed in the kendo matches, suggesting that the complex offensive and defensive maneuvers were generated by simple rules. Furthermore, we detected several functions or patterns during one offensive and defensive scene (Figure 4f), indicating that interpersonal coordination switched among several patterns during the competition.