The purposes of this study were to characterise the total space covered and the distances between players within teams over ten Brazilian First Division Championship matches. Filmed recordings, combined with a tracking system, were used to obtain the trajectories of the players (n = 277), before and after half-time. The team surface area (the area of the convex hull formed by the positions of the players) and spread (the Frobenius norm of the distance-between-player matrix) were calculated as functions of time. A Fast Fourier Transform (FFT) was applied to each time series. The median frequency was then calculated. The results of the surface area time series median frequencies for the first half (0.63 ± 0.10 cycles · min⁻¹) were significantly greater (P < 0.01) than the second-half values (0.47 ± 0.14 cycles · min⁻¹). Similarly, the spread variable median frequencies for the first half (0.60 ± 0.14 cycles · min⁻¹) were significantly greater (P < 0.01) than the second-half values (0.46 ± 0.16 cycles · min⁻¹). The median frequencies allowed the characterisation of the time series oscillations that represent the speed at which players distribute and then compact their team formation during a match. This analysis can provide insights that allow coaches to better control the team organisation on the pitch.
Let M be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of M generated by vector fields in each of of these distributions. Given a stochastic flow ϕ t of diffeomorphisms of M , in a neighbourhood of initial condition, up to a stopping time we decompose ϕ t = ξ t • ψ t where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.
Abstract. We extend the Hartman-Grobman theorems for discrete random dynamical systems (RDS), proved in [7], in two directions: for continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between travelling neighborhoods of trajectories and neighborhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.1. Introduction. The celebrated Hartman-Grobman theorem (HGT, for short) plays a fundamental rule in the theory of dynamical systems. Essentially, among other features, it allows one to make topological classification of the dynamics in a neighborhood of hyperbolic fixed points. This classification is based on the existence of a conjugacy of the local dynamics with the linearized system at a hyperbolic fixed point. For the original papers we mention Hartman [10] and [11], and Grobman [9]. One of the first results concerning HGT in random dynamical systems (RDS, for short) is due to Wanner [19] for discrete systems. His argument was based on random difference equation, such that the construction was made ω-wise. His proof is completed by showing that the choice of random homeomorphisms is, in fact, measurable. In our intrinsic approach, Coayla-Teran and Ruffino [7], we have looked for the conjugacy in an appropriate Banach space of random homeomorphisms. The arguments in [7] correspond to an appropriate extension of the deterministic arguments, with the state space enlarged by the probability space. Hence the norms and other constants were composed with L 1 (Ω) norm. In this article, we extend
We present versions of Hartman-Grobman theorems for random dynamical systems (RDS) in the discrete case. We use the same random norm like in Wanner [14], but instead of using difference equations, we perform an appropriate generalization of the deterministic arguments in an adequate space of measurable homeomorphisms to extend the results in [14] with weaker hypotheses (integrability instead of boundedness) and simpler arguments.
We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φt = ξt°Ψt° Θt, where ξt and Ψt have the same properties of Liao's decomposition and (ξt°Ψt) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξt via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.
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