# A Switching Hybrid Dynamical System: Toward Understanding Complex Interpersonal Behavior

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## Abstract

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## 1. Introduction

## 2. Continuous Dynamics

#### 2.1. The Haken-Kelso-Bunz (HKB) Model

#### 2.2. Identity of Synchronization Modes Revealed by the Relative Phase Region During Interpersonal Competition

## 3. Continuous to Discrete Dynamics: Return Map

#### 3.1. Lorenz Map

#### 3.2. Identification of Coordination Patterns by the Return Map during Interpersonal Competition

#### 3.3. Identification of Switching Pattern for Expertise via the State Transition Probability

## 4. Continuous to Discrete Dynamics: Switching Dynamics

#### 4.1. Switching Dynamics

#### 4.2. Underlying Simple Rule for Complex Striking Actions as per the Poincaré Map

## 5. Switching Hybrid Dynamics

**A**during competition between

**A**and

**B**. The higher module transforms into human movement based on the continuous output pattern from the opponent

**B**, ${I}_{ext}\left(t\right)$ and the final state of the lower module $x\left(t\right)$. The higher module considers one of three patterns ${I}_{l}\left(t\right)$ and transforms into movement. Thus, when the movement pattern of opponent

**B**switches among the three patterns, the movement patterns of

**A**will show three fractal trajectory subsets. However, the pattern determined by

**A**is not always consistent with the continuous output pattern from opponent

**B**, ${I}_{ext}\left(t\right)$. 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 ${I}_{ext}\left(t\right)$ does not always generate the same decision ${I}_{l}\left(t\right)$. This problem might be related to expertise.

**A**and ${x}_{B}\left(t\right)$ in

**B**transforms into external input for the system ${I}_{Bext}\left(t\right)$ in

**B**and ${I}_{Aext}\left(t\right)$ in

**A**, respectively. As a result, the two systems are connected through external inputs. Then, the behavior of the whole system is described as: $\dot{X}=F\left(X\right),X=({x}_{A},{x}_{B})$. 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.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Abrupt changes in the interpersonal coordination pattern corresponding to an interpersonal distance of 2.7–3.0 m while playing kendo. The frequencies of the relative phase per 0.1 m interval at interpersonal distances of 2.7–3.0 m were calculated, and the means and standard deviations are presented. The relative phases were divided into nine ranges (${0}^{\circ}$–${20}^{\circ}$, ${20}^{\circ}$–${40}^{\circ}$, ${40}^{\circ}$–${60}^{\circ}$, ${60}^{\circ}$–${80}^{\circ}$, ${80}^{\circ}$–${100}^{\circ}$, ${100}^{\circ}$–${120}^{\circ}$, ${120}^{\circ}$–${140}^{\circ}$, ${140}^{\circ}$–${160}^{\circ}$, and ${160}^{\circ}$–${180}^{\circ}$). Anti-phase coordination was dominant at interpersonal distances of 2.7–2.8 m. However, in-phase coordination was dominant at interpersonal distances of 2.9–3.0 m. Modified from [34].

**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 $z\left(t\right)$ and peaks of $z\left(t\right)$ are shown as red dots. (

**c**) Lorenz map plotted as ${z}_{n}$ vs. ${z}_{n+1}$.

**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 ${X}_{IPD}\left(t\right)$, normalized ${V}_{IPD}\left(t\right)$, and $X\left(t\right)$, respectively, for a 12-s trial with more than five peaks. The red and gray circles indicate the corresponding values of $X\left(t\right)$ for the peaks of ${X}_{IPD}\left(t\right)$. (

**b**) Return map of the time series for the observed data, ${X}_{n}$ vs. ${X}_{n+1}$ using the amplitude of $X\left(t\right)$ at the peaks of ${X}_{IPD}\left(t\right)$ corresponding to the series of points (red and gray circles) in the panel shown in

**a**. Modified from [37].

**Figure 4.**Examples of a well-fitted series of points by each function using the return map analysis. (

**a**–

**d**) Linear functions, ${X}_{n+1}=a{X}_{n}+b$, with four different slopes for $0<a<1,-1a0,1a$, and $a<-1$, respectively. (

**e**) Exponential function, ${X}_{n+1}=b\phantom{\rule{3.33333pt}{0ex}}exp(a{X}_{n})$, and logarithmic function, ${X}_{n+1}=alog({X}_{n})+b$. (

**f**) Examples of switching functions in one scene. The red lines show attractors, blue lines show repellers, and cyan lines show intermittencies. Modified from [37].

**Figure 5.**Second-order state transition diagrams with conditional probabilities consisting of the “farthest apart” high velocity states (F) and the “nearest together” low velocity state (N) for expert (

**a**) and intermediate (

**b**) competitors, respectively. Modified from [37].

**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 ${\overline{x}}_{1},{\overline{x}}_{2}$ and ${\overline{x}}_{3}$, 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.

**Figure 8.**Examples of Poincaré sections $\Sigma $ for periodic and switching input conditions. (

**a**) Periodic input, (

**b**) switching input, and (

**c**) the ellipse of constant distance using each mean and $\pm 1\phantom{\rule{0.277778em}{0ex}}S.D.$ for switching input. (

**d**) The return map shows the construction of the Cantor set with rotation using two iterative functions. The iterative functions ${g}_{F}$ and ${g}_{B}$ transform the state, ${x}_{\tau}$, to the next state, ${x}_{\tau +1}$. The transformations of the iterative functions ${g}_{F}$ and ${g}_{B}$ are rotated around the fixed points ${x}_{F}$ and ${x}_{B}$, respectively. (

**e**) The hierarchical structure of the fractal corresponds to the sequence of forehand (F) and backhand (B) inputs. Modified with permission from [40], Hum. Mov. Sci. 2000.

**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.

**Figure 10.**Schematic representation for two-coupled switching hybrid dynamics. This system is autonomous.

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**MDPI and ACS Style**

Yamamoto, Y.; Kijima, A.; Okumura, M.; Yokoyama, K.; Gohara, K.
A Switching Hybrid Dynamical System: Toward Understanding Complex Interpersonal Behavior. *Appl. Sci.* **2019**, *9*, 39.
https://doi.org/10.3390/app9010039

**AMA Style**

Yamamoto Y, Kijima A, Okumura M, Yokoyama K, Gohara K.
A Switching Hybrid Dynamical System: Toward Understanding Complex Interpersonal Behavior. *Applied Sciences*. 2019; 9(1):39.
https://doi.org/10.3390/app9010039

**Chicago/Turabian Style**

Yamamoto, Yuji, Akifumi Kijima, Motoki Okumura, Keiko Yokoyama, and Kazutoshi Gohara.
2019. "A Switching Hybrid Dynamical System: Toward Understanding Complex Interpersonal Behavior" *Applied Sciences* 9, no. 1: 39.
https://doi.org/10.3390/app9010039