Dynamical Systems Excited by Temporal Inputs: Fractal Transition Between Excited Attractors

Gohara, Kazutoshi; Okuyama, Arata

doi:10.1142/s0218348x99000220

Cited by 28 publications

(37 citation statements)

References 14 publications

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“…Furthermore, we are able to estimate the relationship between the set C or the set of trajectories Γ(C) and the switchingtime length T [Gohara & Okuyama, 1999a]. T is also the duration of the external input.…”

Section: Dynamical System Excited By Switching Inputsmentioning

confidence: 99%

“…a control parameter. In contrast, a dynamical system, which is open to other systems, is considered a nonautonomous system or a nonautonomous dynamical system [Gohara & Okuyama, 1999a]. In this system, movement patterns are excited by external inputs that are external parameters.…”

Section: Introductionmentioning

confidence: 99%

“…We need to adopt the latter approach and the corresponding theory to examine this dynamics in order to study human motor control, and the difference between this study and the studies by Kelso and his colleagues is that the experiment in our study is based on the nonautonomous dynamical systems theory. The first study in which the dynamical systems theory [Gohara & Okuyama, 1999a] involving switching stochastically with several external inputs was applied to human movement was carried out by Yamamoto and Gohara [2000]. They focused on the periodic movement patterns of forehand and backhand strokes of tennis players as attractors were excited with external inputs.…”

Section: Introductionmentioning

confidence: 99%

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Inverse Proportional Relationship Between Switching-Time Length and Fractal-Like Structure for Continuous Tracking Movement

Hirakawa

Suzuki

Gohara

et al. 2017

Int. J. Bifurcation Chaos

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We investigate the relationship between the switching-time length T and the fractal-like feature that characterizes the behavior of dissipative dynamical systems excited by external temporal inputs for tracking movement. Seven healthy right-handed male participants were asked to continuously track light-emitting diodes that were located on the right and left sides in front of them. These movements were performed under two conditions: when the same input pattern was repeated (the periodic-input condition) and when two different input patterns were switched stochastically (the switching-input condition). The repeated time lengths of input patterns during these conditions were 2.00, 1.00, 0.75, 0.50, 0.35, and 0.25 s. The movements of a lever held between a participant's thumb and index finger were measured by a motion-capture system and were analyzed with respect to position and velocity. The condition in which the same input was repeated revealed that two different stable trajectories existed in a cylindrical state space, while the condition in which the inputs were switched induced transitions between these two trajectories. These two different trajectories were considered as excited attractors. The transitions between the two excited attractors produced eight trajectories; they were then characterized by a fractal-like feature as a third-order sequence effect. Moreover, correlation dimensions, which are typically used to evaluate fractal-like features, calculated from the set on the Poincaré section increased as the switching-time length T decreased. These results suggest that an inverse T. Hirakawa et al. proportional relationship exists between the switching-time length T and the fractal-like feature of human movement.

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Section: Dynamical System Excited By Switching Inputsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inverse Proportional Relationship Between Switching-Time Length and Fractal-Like Structure for Continuous Tracking Movement

Hirakawa

Suzuki

Gohara

et al. 2017

Int. J. Bifurcation Chaos

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“…In Figure 3, the horizontal axis shows the shoulder segment angle (x 1 ) and the vertical axis the hip segment angle (x segment angle (x segment angle ( 2 ) in the discrete dynamical system in conventional non-linear dynamics into the continuous dynamical system [Gohara & Okuyama, (1999)]. …”

Section: Striking Action Dynamicsmentioning

confidence: 99%

An Alternative Approach to the Acquisition of a Complex Motor Skill: Multiple Movement Training on Tennis Strokes

Yamamoto¹

2004

Int. J. Sport Health Sci.

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This study examined an alternative approach to the acquisition of a complex motor skill, using constraints. To determine the effi cacy of two alternative methods of learning tennis strokes, forehand and backhand strokes were practiced over fi ve successive days using two contrasting training methods. The novel, multiple movement method required alternate forehand and backhand strokes, whereas the traditional, simple movement method required repeated forehand or backhand strokes as separate series of events. Five subjects were randomly assigned to each of the two conditions. Their strokes were recorded and analyzed kinematically before and after the training sessions. The simple movement method resulted in no changes in the range of trunk rotation perse, but the center of the range shifted unfavorably, in the direction opposite to that in which the ball was struck, for both forehand and backhand strokes. By contrast, the multiple movement method increased the range of trunk rotation, and the center of the range shifted favorably in the direction toward which the ball was struck, for both strokes. These differences were confi rmed by the trajectories observed in hyper-cylindrical phase space as dynamical systems. In the multiple movement group, the forehand and backhand clusters converged after training, whereas in the simple movement group the two clusters diverged after training. From a dynamical systems perspective, we argue that the multiple movement method achieves its superiority by exploiting the inertia of the trunk rotation movement as a constraint that is produced by the preceding striking action, whether it be forehand or backhand.

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“…Buchanan and Kelso, 1993Carson et al, 1995Schöner et al, 1986, Gohara and Okuyama, 1999aGohara and Okuyama, 1999aGohara et al, 1999cNishikawa and Gohara, 2008 1 Wilks λ f 2 Table 3 Table 4 Figure 5 Figure 5 Expert Novice (d) show the Poincaré maps for switching input condition. A black circle ( ) indicates that the preceding and current input was foreside, and a white circle ( ) indicates that the preceding input was backside and the current one was foreside.…”

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confidence: 99%

Dexterity of Switched Hitting Movement Using Fractal Dimension Analysis

Suzuki¹,

Yamamoto²

2013

JJSP

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The purpose of this study was to quantify human dexterity by examining the movement involved in switching between forehand and backhand strokes when a ball moved from side to the other during table tennis. The hitting movements of expert and novice table tennis players were observed when balls were repeatedly moving in the same direction (periodic input condition) and when they were moving in two different directions successively (switching input condition). From the viewpoint of the switching dynamical system (Gohara and Okuyama, 1999a), the repeated movement under the periodic input condition was treated as an attractor, and the switching movement between strokes under the switching input condition was treated as transition of attractors. The dexterity with which movement were completed was quantified in terms of the fractal dimension. The fractal dimension was calculated according to Poincaré maps depicting the trajectories of the midpoint and angular velocities at the shoulder. Data from experts and novices almost reflected transitions of the third-order sequence effect, and the fractal dimensions included non-integers, which indicate that these fractal transitions had fractal properties. However, the fractal dimension of experts was lower than that of novices. The two output patterns corresponding to the two input patterns overlapped more for novices than for experts.The results suggest that the dexterity shown in switching movements can be quantified in terms of the fractal dimension based on the switching dynamical system.

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Dynamical Systems Excited by Temporal Inputs: Fractal Transition Between Excited Attractors

Cited by 28 publications

References 14 publications

Inverse Proportional Relationship Between Switching-Time Length and Fractal-Like Structure for Continuous Tracking Movement

Inverse Proportional Relationship Between Switching-Time Length and Fractal-Like Structure for Continuous Tracking Movement

An Alternative Approach to the Acquisition of a Complex Motor Skill: Multiple Movement Training on Tennis Strokes

Dexterity of Switched Hitting Movement Using Fractal Dimension Analysis

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