## 1 Introduction  
Exploring complex human behavior from the perspective of a dynamical system began with a historic experiment [1]. Three decades later, a new approach was developed using dynamical system theory and utilities to analyze and simulate complex phenomena, such as human movement, using super-computers. In this article, we review the applications of a dynamical system approach to human behavior, with particular regard to sports activities that require quick decision-making and appropriate execution. We provide applications for continuous dynamics, discrete dynamics, switching dynamics, and switching hybrid dynamics, with a short theoretical review.  
Human behavior, particularly the behavior of rhythmic cyclical movements, can be understood as continuous dynamics that can be described using differential equations. In 1665, the Dutch physicist [PERSON] synchronized two pendulum clocks using the interactions of tiny vibrations in their common support [2; 3]. Synchronizing the two clocks was regarded as entrainment and/or synchronization of the two coupled nonlinear oscillators model. The theoretical Haken-Kelso-Bunz (HKB) model describes the phase transition during the coordinating pattern observed between hands based on synergetics [4] and the nonlinear oscillator theory [5]. This model has been applied to intrapersonal and interpersonal coordination [6; 7].  
Furthermore, group synchrony or coordination has been examined using the Kuramoto model and/or the Kuramoto order parameter [8], with regard to synchronized clapping [9] and phase synchronization among rocking chairs in a small group [10; 11]. Anti-phase synchronization of the tree frogs was analyzed as a phase oscillator model [12]. As another approach, the symmetric Hopf bifurcation theory based on group theory [13] was applied to investigate the synchronized pattern of three people during a sports activity [14]. These approaches could be regarded as oscillator dynamics, or continuous dynamics.  
On the other hand, discrete dynamics and/or mapping can be depicted as iterated maps, that is, in terms of difference equations describing the time evolution, such as population dynamics with generations. The logistic map is a well-known map of how complex, chaotic behavior can arise from very simple quadratic difference equations. A Markov chain in discrete time, which is characterized as a state transition based on the probability distribution of the next state, depends only on the current state as a kind of discrete dynamics, and has been applied to baseball batting [15] and to squash [16; 17]. However, these studies examined predictions pertaining to competitive sports performance and did not explore the underlying dynamics. We introduce examples to explore the simple rules underlying complex human behavior using a Lorenz map, that is, a first return map and a Poincare map to convert continuous dynamics to discrete dynamics.  
Previous studies have been based on the two coupled nonlinear oscillators model. Thus, these systems are closed systems with no abrupt changes in external input. We introduce the application of switching dynamics with temporal input to continuous striking movements. This model also uses the idea of the Poincare map. Finally, to explore complex human movement as a dynamical system, we introduce switching hybrid dynamical systems, which include discrete and continuous dynamics with a feedback loop.