Structured Flows on Manifolds: A short introduction & applications to sports

Structured Flows on Manifolds: A short introduction & applications to sports

Resumen

The recognition that motor behaviour may well constitute a self-organized pattern formation process has been steadily growing, resulting in an increased application of concepts and tools from dynamical systems theory and theories of self-organization in motor control research and related domains. In the sport sciences, however, this appears less so, maybe to some extent because actions in sports are typically complex (i.e., high-dimensional) while the dynamical approaches have largely focussed on rather simple movements. Here, we introduce a general dynamical framework, Structured Flows on Manifolds (SFM), for the control of movements (or processes, more in general) of arbitrary complexity. Phase flows in state (or phase) space, to be discussed in-depth by Perdikis, are key to our framework. A system’s dynamics is composed of a fast dynamics that quickly collapses onto a so-called manifold (a subspace of the phase space) and a slower low-dimensional flow onto it. The time evolutions of high-dimensional systems generally co-vary considerably and are effectively described by the low-dimensional flow. These processes will be presented in a mathematical formalism and illustrated via numerical simulations. Subsequently, we show how the low-dimensional dynamics can be extracted from experimental data using principal component analysis (PCA). PCA is a tool that aims for an effective representation of high-dimensional data based on their covariance, allowing for the separation of dynamics that are ‘relevant’ and those that are less so. In combination, SFM and PCA provide the sports sciences with a conceptual toolbox to address the inherently complex phenomena of its interests.