Abstract
We investigate the relationship between the switching-time length [Formula: see text] 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[Formula: see text]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 [Formula: see text] decreased. These results suggest that an inverse proportional relationship exists between the switching-time length [Formula: see text] and the fractal-like feature of human movement.
Highlights
The famous early works in which dynamical systems theory was applied to human movement were the experimental verification by Kelso [1984] and the mathematical model of Haken et al [1985]
We show the relationship between Eq (2) as the discrete dynamical system and Eq (3) as the continuous dynamical system from the perspective of the nonautonomous dynamical system by analyzing the set on the Poincare section Σ in the hyper-cylindrical state space M
The trajectories before and after the Poincare section Σ(θ = 0, 2π) were omitted to confirm the set C on the section more. These results show that the trajectories of output patterns, which were excited by the external inputs L and R, were different in the continuous dynamical system
Summary
The famous early works in which dynamical systems theory was applied to human movement were the experimental verification by Kelso [1984] and the mathematical model of Haken et al [1985]. Recent studies based on this dynamical systems theory, for example the physical phenomenon [Arecchi et al, 1986; Constantin et al, 1991; Maas et al, 1997; Matias et al, 1997; Mestl et al, 1997; Schmiegel & Eckhardt, 1997; Tanii et al, 1999; Tanii et al, 1991] and human movement [Hirakawa et al, 2016; Suzuki & Yamamoto, 2015; Yamamoto & Gohara, 2000], have revealed that the spatiotemporal structure in various natural phenomena is fractal-like and/or self-similar These studies on human movement entailed tasks in which participants switched several different movement patterns continuously with the external inputs being abruptly switched
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