Super Attractor: Methods for Manifesting a Life beyond Your Wildest Dreams

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Similarity rate of running measurements (triangle pointing right) and simulations (triangle pointing left). Dingwell JB, Cusumano JP. Nonlinear time series analysis of normal and pathological human walking. Chaos. 2000;10(4):848–63. pmid:12779434

Let t {\displaystyle t} represent time and let f ( t , ⋅ ) {\displaystyle f(t,\cdot )} be a function which specifies the dynamics of the system. That is, if a {\displaystyle a} is a point in an n {\displaystyle n} -dimensional phase space, representing the initial state of the system, then f ( 0 , a ) = a {\displaystyle f(0,a)=a} and, for a positive value of t {\displaystyle t} , f ( t , a ) {\displaystyle f(t,a)} is the result of the evolution of this state after t {\displaystyle t} units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R 2 {\displaystyle \mathbb {R} Wilson RC, Jones PW. A comparison of the visual analogue scale and modified Borg scale for the measurement of dyspnoea during exercise. Clinical Science. 1989;76(3):277–82. pmid:2924519 The transient effect is a temporary oscillation around the attractor at the beginning of a cyclic movement. The starting value of the oscillation might be very individual, specific to the subject, and having a part of the starting value occurring by sheer chance. We model the deviation as the solution of a damped harmonic oscillator, where the transient term can be viewed as the departure from the morphed attractor Hausdorff JM, Zemany L, Peng C, Goldberger AL. Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J Appl Physiol (1985). 1999;86(3):1040–7. at attractor point j. Here b is the controlling constant and σ k( j) the attractor’s standard deviation, which is divided by the average of the attractor’s deviation 〈 σ k〉. This takes care of the changing width of the acceleration bundle. The correction term, being activated at time t b, is modeled as

Introduction

Buchthal F, Schmalbruch H. Contraction Times and Fibre Types in Intact Human Muscle. 1970;79(4):435–52. Bramble DM, Lieberman DE. Endurance running and the evolution of Homo. Nature. 2004;432(7015):345–52. pmid:15549097 Strogatz SH. Nonlinear dynamics and Chaos: with applications to physics, biology, chemistry, and engineering. Reading, Mass.: Addison-Wesley Pub.; 1994. xi, 498 p. p. Bipedal gait, especially walking, has been the most decisive development of homo sapiens to surpass their ancestors and relatives [ 1]. In the past centuries further cyclic motions like swimming, cycling, rowing or skiing came along, to overcome natural obstacles, to facilitate traveling and then as leisure activities. Recently, cyclic motion descriptions have served as biological templates for developments in robotics together with developments in artificial intelligence [ 2]. Although cyclic movements are performed a thousand-fold each day in everyday life, their underlying composition and structure is not fully understood.

Visual representation of a strange attractor. [1] Another visualization of the same 3D attractor is this video. Code capable of rendering this is available.England SA, Granata KP. The influence of gait speed on local dynamic stability of walking. Gait Posture. 2007;25(2):172–8. pmid:16621565 There exist two types of models describing human cyclic motion—theory driven and data driven [ 18]–both with its own strong and weak aspects. For example, a theory driven model as described by Gerritsen et al. [ 19] gives insight into the working of seven muscle groups within the lower extremities. The necessity of keeping the model manageable, in the mentioned paper by using a 2-dimensional rigid body model, leads to deviations from the actual movement. On the other hand the data driven model of Janssen et al. [ 18] was able to detect the influence of emotions onto the movement pattern. They applied deep machine learning by using artificial neural nets, allowing identification of subtle effects. While here the detection movement characteristics caused by emotions is nicely achieved, the specifics of the gait changes remained undetected. With the present paper we attempt a compromise, by not having to rely on anatomy and muscle function, but still trying to understand kinematic processes and the movement pattern quantitatively. A study on cycling at two different power outputs (150 W and 300 W) at a cadence of 90 rpm [ 20] found differences in the muscle activities detected via EMG, while kinematic data stayed almost unchanged. This result together with the stability of the individual’s attractor over time and after rehabilitation [ 21, 22] is motivation to examine the possibility to quantitatively describe movement without the knowledge of muscle activity. Barnes KR, Kilding AE. Strategies to improve running economy. Sports Medicine. 2015(45(1)):37–56. pmid:25164465 With sign(…)being the signum and Θ(…)the step function. We set the maximal acceleration change to τ = 80 ms analogous to the style of a muscle’s timely response [ 24] with acceleration effectively lasting t M = 4⋅ τ = 320 ms, to obtain

Clermont CA, Benson LC, Osis ST, Kobsar D, Ferber R. Running patterns for male and female competitive and recreational runners based on accelerometer data. Journal of sports sciences. 2019;37(2):204–11. pmid:29920155 Weich C, Jensen RL, Vieten M. Triathlon transition study: quantifying differences in running movement pattern and precision after bike-run transition. Sports Biomech. 2019;18(2):215–28. pmid:29141506

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Loske S, Nuesch C, Byrnes KS, Fiebig O, Scharen S, Mundermann A, et al. Decompression surgery improves gait quality in patients with symptomatic lumbar spinal stenosis. Spine J. 2018. RU[ α, β] represents random generation with a uniform characteristic within the interval [ α, β]. With this definition the standard deviation of the random walk depends on the sampling frequency f S. Since the random walk must not be dependent on the specifics of a measurement–the sampling frequency f S -, we introduce a parameter ϕ (random walk’s strength), which does not change with the sampling frequency. Al-Zahrani KS, Bakheit MO. A historical review of gait analysis Reply from the Author. Neurosciences. 2008;13(4):460-. pmid:21063384 RN[1, σ M]( t) represents a normally distributed random element introducing some deviation from a perfect working controlling mechanism.