## 4 Continuous to Discrete Dynamics: Switching Dynamics
### Underlying Simple Rule for Complex Striking Actions as per the Poincare Map  
We applied these switching dynamics to the striking action during tennis to understand complex human movements [40]. Two kinds of trajectories occurred when the ball was launched to the forehand or backhand side repeatedly as a periodic input condition. The trajectories were termed excited attractors, according to the input type (Figure 7a). When the ball was launched to the forehand and backhand sides randomly as a switching input condition, the trajectories in the cylindrical phase space were more complex (Figure 7b).  
Figure 6: (**a**) Examples of time series for three periodic inputs. \(I\) and \(O\) denote the input and output time series, respectively. The trajectories for three periodic inputs in three-dimensional cylindrical phase space, \((x_{1},x_{2},\theta)\in\mathcal{M}:R^{2}\times S^{1}\), corresponding to the colored trajectories denoted by \(A_{1}\), \(A_{2}\) and \(A_{3}\) cross the Poincaré section \(\Sigma:R^{2}\) at \(\tau_{1}\), \(\tau_{2}\) and \(\tau_{3}\), respectively. (**b**) An example of a time series for switching inputs. (**c**) The trajectories of randomly switching inputs and the cross points on the Poincaré section show the Sierpinski gasket as a result of the fractal transitions. Modified with permission from [39], Fractals 1999.  
Figure 7: Trajectories in three-dimensional cylindrical phase space. (**a**) Periodic input condition, (**b**) switching input condition. The stick pictures show forehand and backhand striking movements at each point in the time series. Modified with permission from [40], Hum. Mov. Sci. 2000.  
To understand the behavior of the system, we examined the Poincare map on the Poincare section, \(\Sigma\), as discrete dynamics. Figure 8a shows the set of points on the Poincare section under a periodic input condition corresponding to Figure 7a. Figure 8b shows the set of points on the Poincare section during a switching input condition corresponding to the Figure 7b, and Figure 8c shows the ellipse of the constant distance by using the mean and \(\pm 1\)\(S.D.\) for the switching input condition. The characteristic configurations of the four clusters of sets on the Poincare sections, that is, BF, FF, BB and FB from the top, corresponded to the Cantor set with rotation.  
Figure 8d shows the return map of the Cantor set, with rotation leaving from the initial state, \(x_{0}\), when the first input was the backhand side, the next state, \(x_{1B}\), and returned to B. When the second input was the backhand side, to the next state, \(x_{2B}\), returned to BB. In contrast, when the second input was the forehand side, the next state, \(x_{2F}\), returned to BF. As a result, we could obtain the characteristic configurations of the four clusters at \(x_{2}\) in Figure 8e corresponding to the empirical data. The behavior of the system was understood as the time evolution of the Cantor set with rotation (Figure 8e). In other words, the time evolution of the striking action corresponding to the two input types was characterized as a fractal transition of the Cantor set. These findings suggest that a Poincare map analysis reveals the underlying simple rule related to exploitation of inertia in complex striking action.