## 5 Switching Hybrid Dynamics
In addition, external input feeds into the higher module, as shown in Figure 9.  
Discrete dynamical system: \[I_{l}(t);l = \sigma(I_{ext}(t),x(t))\] (4) Continuous dynamical system: \[\dot{x} = f_{l}(x(t),I_{l}(t))\] (5)  
Here, the discrete dynamical system \(I_{l}(t)\) is the higher module corresponding to the brain and prefrontal cortex [42; 43], and the continuous dynamical system \(\dot{x}\) is the lower module corresponding to the human motor system. This system focuses on individual \(\mathbf{A}\) during competition between \(\mathbf{A}\) and \(\mathbf{B}\). The higher module transforms into human movement based on the continuous output pattern from the opponent \(\mathbf{B}\), \(I_{ext}(t)\) and the final state of the lower module \(x(t)\). The higher module considers one of three patterns \(I_{l}(t)\) and transforms into movement. Thus, when the movement pattern of opponent \(\mathbf{B}\) switches among the three patterns, the movement patterns of \(\mathbf{A}\) will show three fractal trajectory subsets. However, the pattern determined by \(\mathbf{A}\) is not always consistent with the continuous output pattern from opponent \(\mathbf{B}\), \(I_{ext}(t)\). Because higher brain functions, such as selective attention [44; 45; 46; 47], visual search strategy [48; 49; 50], and decision making [51; 52; 53] are redundant due to neuronal redundancy of cell assemblies [54], the same external input \(I_{ext}(t)\) does not always generate the same decision \(I_{l}(t)\). This problem might be related to expertise. When the external inputs feed into the system, system outputs of the continuous dynamical system are generated regularly depending on continuous switching of external inputs \(I_{ext}(t)\) and the final states of the system \(x(t)\), as shown in Section 4.2. This suggests that the behavior of the system shows hysteresis and can be used to predict the next state. However, it has been confirmed that the regularities differ according to the length of time of the external input [55; 56; 57]. In this case, we focused on individual behavior during an interpersonal competitive situation; that is, we regarded the system as non-autonomous. When these two systems are connected to each other, it is regarded as an autonomous system as a whole (Figure 10). That is, the final state of the other system, \(x_{A}(t)\) in **A** and \(x_{B}(t)\) in **B** transforms into external input for the system \(I_{Bext}(t)\) in **B** and \(I_{Aext}(t)\) in **A**, respectively. As a result, the two systems are connected through external inputs. Then, the behavior of the whole system is described as: \(\dot{X}=F(X)\), \(X=(x_{A},x_{B})\). In the case of kendo matches, the behavior of the whole system has been described as the instantaneous relative phase difference of the step toward-away movements of the two players. However, six offensive and defensive maneuver patterns have been found, and these patterns switch continuously during a kendo match, suggesting that the regularity underlying switching among competitive patterns could be clarified if these patterns are regarded as output patterns and/or external input patterns. To this end, we need longer time windows to observe switching among combinations of several patterns, compared to that needed for clarifying each pattern. However, the regularity remained unclear after determining the state transition probabilities for the competitive patterns. Applying switching hybrid dynamics to interpersonal competitive behavior would help to clarify how the behavior will be generated during the time course as a whole system. Figure 9: Schematic representation of switching hybrid dynamics, which is composed of a discrete dynamical system as a higher module and a continuous dynamical system as a lower module with a feedback loop. This system is non-autonomous.