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Edited by: Costantino Balestra, Université Libre De Bruxelles, Belgium

Reviewed by: Ruud J. R. Den Hartigh, University of Groningen, Netherlands; Angelo De Santis, University of Chieti-Pescara, Italy

*Correspondence: Tiago M. Barbosa

This article was submitted to Movement Science and Sport Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The aim of this study was to compare the non-linear properties of the four competitive swim strokes. Sixty-eight swimmers performed a set of maximal 4 × 25 m using the four competitive swim strokes. The hip's speed-data as a function of time was collected with a speedo-meter. The speed fluctuation (

Water is a unique and challenging environment for humans who are not specially prepared to propel themselves in this environment. Competitive swimmers use one of the four swim strokes as locomotion technique. Swimming is a periodically accelerated motion (Barbosa et al., _{0} is the subject's velocity at the beginning of the stroke cycle, Δ_{i} = _{0}). The mechanism underlying the accelerations and decelerations (or speed fluctuation) within each stroke is related to two external forces (propulsive force and hydrodynamic drag) acting upon the swimmer and it is an application of Newton's law of motion:
_{Pr} is the total propulsive force (in the traveling direction of displacement), _{D} is the hydrodynamic drag force (opposite to the traveling direction), and _{Pr} higher than _{D}, while negative slopes correspond to _{Pr} lower than _{D}. One part of the _{Pr} produces mechanical work to overcome _{D} (_{D} = ^{.} ^{2}), so:
_{Fd} is the mechanical work, ^{.} ρ^{.} S ^{.} _{D}; fluid density, surface area and drag coefficient, respectively), _{0} is the subject's velocity at the beginning of the stroke cycle, Δ

There is a solid body of knowledge describing speed fluctuation in human swimming by the coefficient of variation (

In a linear system, a small change in one input has a proportional and quantifiable change in the output. As far as human movement concerns (and notably in elite sports), this may not always be the case. Sometimes small changes in the input are not reflected in the variables selected to monitor one's motor behavior. In such event, non-linear parameters are quite useful because they exhibit a very sensitive dependence on the inputs. Non-linear complex dynamical systems are characterized by interaction-dominant dynamics, which is at odds with component dominance and with additive effects. In elite sports, practitioners bridge these concepts to the marginal gains “theory.” The latter encompasses the rationale that it is the sum of very small changes (each one of them might be non-significant) that helps the elite athlete to excel. It is hypothesized that such small changes can be monitored by non-linear parameters. In the motor control of a biological system, the variables playing a role on a main outcome are not independent. There is an interplay among several variables that ultimately will affect the main outcome. Therefore, one may reason that each marginal gain will trigger a change in the interplay among the components of the system affecting ultimately the main outcome. Under complex science it is more accurate to note that rather than the sum of trivial changes, it is the dynamic interaction in play that may help to excel. Each trivial change that a practitioner may point out might indeed be a change in the dynamic interaction of the systems' components though.

Academics with research interest on these topics, note that the constraints-led approach is an interesting framework to be considered (Davids et al.,

The entropy is an informational non-linear parameter that describes the degree of irregularity/complexity inherent to the order of the elements in a time-series (Bravi et al.,

Fractal dimension is categorized as an invariant non-linear parameter describing the properties of a system that demonstrates fractality or other properties that do not change over time and/or space (Bravi et al.,

The aim of this research was to compare the non-linear properties of the four swim strokes. It was hypothesized that like other locomotion techniques, swimming will exhibit non-linear proprieties, including the entropy and fractal dynamics. However, because of the different configurations of constraints acting on the swim strokes, there will be differences among the four.

Sixty-eight high-level swimmers were assessed (34 males: 17.06 ± 4.11 years old; 34 females: 14.97 ± 2.96 years old). The sample included age-group national record holders, age-group national champions, and other swimmers that compete on regular basis at national or international competitions.

Coaches, parents or guardians, and the swimmers gave informed written consent/assent for participation in this study. All procedures were in accordance with the Helsinki Declaration regarding human research. The University IRB also approved the research design.

The swimmers did a standard warm-up of 1500 m including continuous swimming at low-moderate intensity, with specific drills and sprints at the end. Each swimmer undertook a set of all-out (i.e., maximal bouts) 4 × 25 m swims using randomly assigned Front-crawl or Backstroke or Breaststroke or Butterfly strokes. Swims started with a push-off and there was a 30 min rest between trials. Participants performed each trial alone with no other swimmer in the lane or nearby lanes to reduce drafting and pacing effects, and extra drag force due to exogenous factors. The swimmers were advised to start swimming after the push-off, minimize the gliding, and dolphin kicking.

A speedo-meter cord (Swim speedo-meter, Swimsportec, Hildesheim, Germany) was attached to the swimmer's hip (Barbosa et al.,

The intra-cyclic variation of the horizontal velocity of the hip (i.e.,

Where _{i} is the instant swimming velocity, _{i} is the acquisition frequency, and

The _{im} is the fraction of patterns of length, _{im} is the number of patterns that are similar between two sets (given the similarity criterion,

There are two main algorithms reported in the literature to compute the

Data normality was tested by the Shapiro–Wilk test. Data is described as mean ± 1

Repeated measures (within-subjects' ANOVA) analysis was performed to compare the four swimming strokes (^{2}) and interpreted as: Without effect if 0 < η^{2} ≤ 0.04; minimum if 0.04 < η^{2} ≤ 0.25; moderate if 0.25 < η^{2} ≤ 0.64 and; strong if η^{2}> 0.64. Cohen's

Analysis across the four strokes returned significant variations with moderate-strong effects in all variables [_{(3, 201)} = 596.498, ^{2} = 0.89; _{(3, 201)} = 89.074, ^{2} = 0.57; _{(3, 201)} = 61.112, ^{2} = 0.47] (Table

^{2} |
|||||||
---|---|---|---|---|---|---|---|

14.04 ± 4.65 |
13.44 ± 3.42 |
39.72 ± 4.81 |
25.62 ± 4.16 |
596.498 | <0.001 | 0.89 | |

0.68 ± 0.15 |
1.03 ± 0.19 |
0.73 ± 0.10 |
0.85 ± 0.17 |
89.074 | <0.001 | 0.57 | |

1.84 ± 0.08 |
1.82 ± 0.07 |
1.92 ± 0.03 |
1.88 ± 0.07 |
61.112 | <0.001 | 0.47 |

To examine the effects of sex and speed as potential confounding and interacting factors, multivariate analysis was computed (Table ^{2} ≤ 0.34). Hence, the sex and the range of speeds swam did not have an effect on the data.

_{Pillai's} |
_{Wilk's} |
^{2} |
_{Pillai's} |
_{Wilk's} |
^{2} |
_{Pillai's} |
_{Wilk's} |
^{2} |
|||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.069 | 0.931 | 1.553 | 0.21 | 0.010 | 0.040 | 0.960 | 0.884 | 0.45 | 0.005 | 0.109 | 0.893 | 1.233 | 0.30 | 0.016 | |

0.052 | 0.948 | 1.151 | 0.34 | 0.007 | 0.061 | 0.939 | 1.370 | 0.26 | 0.010 | 0.096 | 0.906 | 1.07 | 0.38 | 0.015 | |

0.023 | 0.977 | 0.484 | 0.69 | 0.008 | 0.053 | 0.947 | 1.164 | 0.33 | 0.010 | 0.096 | 0.906 | 1.07 | 0.38 | 0.034 |

The aim of this study was to compare the non-linear properties of the four competitive swim strokes. As hypothesized, swimming does exhibit non-linear properties that are different among the four swimming strokes (0.68 ≤

The

The selection of multiple non-linear measures (for this research the

The value of

The degree of complexity exhibited by each swim stroke can depend on several constraints experienced by the subjects. The constraint-led approach by Davids et al. (

This study provides new insights into swim analysis, whereby non-linear properties differentiate the four swim-strokes. These results encourage the use of non-linear properties to analyze swimming beyond the traditional methods. Future research on this topic should focus on examining: (i) if these non-linear properties change according to the expertise level of the subjects recruited; (ii) if human swimming does exhibit non-linear properties, future research should focus on the understanding of the mechanisms underpinning such phenomenon (e.g., how the constraints-led perspective or a model of self-organized criticality, interaction dominant dynamics, or degeneracy can explain the complexity of the motor behavior).

It can be concluded that swimming data exhibits non-linear properties, which are different among the four competitive swimming strokes. The

Conceived and designed the experiments: TB, MC, DP. Performed the experiments: WG. Analyzed the data: TB, WG, JM, Draft the manuscript: TB, WG, JM, MC, DP.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to express their deep gratitude to Mr. Lim Aik Ho and Mr. Huang Wei Lun for helping with data collection. This research was funded by the NIE AcRF grant (RI 11/13 TB).