Dynamical Systems

Every biological system—microbe, athlete, or team—can be described formally in terms of its time evolution or dynamics. This becomes feasible if one capitalizes on a crucial characteristic of biological systems, namely that they exchange energy and matter, and in some cases also information, with their surroundings. In statistical physics, such systems are called open systems, as opposed to closed systems without environmental contact. Certain classes of open systems may self-organize and form coherent macroscopic patterns due to interactions between their constituent subsystems. When applied to sports behavior, the term subsystems may refer to the neurons, muscles, organs, and limbs of an individual athlete, but also to the individual athletes themselves, for instance when in combat with an opponent, be it one on one (boxing), one against many (cycling), or in teams (football). Furthermore, the term system may refer to virtually any aspect of sports behavior, including action, perception, emotion, and cognition, as well as their physiological underpinnings. Finally, each of these behaviors may change as a function of development and learning, which introduces an extra layer of dynamics. All of this implies that the opportunities for applying the concepts and methods of dynamical systems theory in sports science are virtually boundless and that any attempt at a priori demarcation is futile. This entry first provides an intuitive, yet necessarily abstract, introduction to the main concepts of dynamical systems theory, which are then brought alive by a collection of examples that highlight their relevance to sports science.

Concepts: A Fling With Physics and Mathematics

An important feature of open systems consisting of many interacting subsystems is that they form spatial, temporal, or functional behavioral patterns. Such coherent macroscopic patterns may be described by a small number of collective variables, called order parameters. Spontaneous switches between macroscopic patterns are termed non-equilibrium phase transitions1 in equivalence with similar qualitative, structural changes studied in statistical physics. Phase transitions may be induced by continuous, gradual changes in relevant system parameters that do not specify the macroscopic patterns and are called control parameters. In what follows, it will be argued that, in the vicinity of these phase transitions, any system can, to a good approximation, be described as a low-dimensional dynamical system. The art of dynamical systems modeling is to find appropriate low-dimensional descriptors—that is, the order parameters—and to derive dynamical equations that describe their time-evolution and nonlinear dependence on control parameters. Such equations provide formal analogies of the macroscopic patterns and phase transitions exhibited by the system under study.

Steady States: Fixed Points, Limit Cycles, and More

Like macroscopic behaviors, dynamical equations often show steady-state behavior. That is, as time goes to infinity, they will settle asymptotically on a steady-state solution, which is often an attractor (a subspace of the state space to which trajectories are attracted). Dynamical systems theory may display four types of attractors: fixed points, limit cycles, limit tori, and chaotic attractors. Fixed points and limit cycles are the simplest attractors: a single point or a series of connected points that are visited at certain intervals, respectively. Some attractors are governed by more than one frequency. If these frequencies stand in rational relation to one another (are commensurate), the resulting behavior will still be a limit cycle. However, if these frequencies stand in irrational relation to one another (are incommensurate), the resulting trajectory is no longer closed, and the limit cycle becomes a limit torus. In this case, the resulting behavior is called quasiperiodic, implying that the trajectory passes arbitrarily close to every point on the torus without ever revisiting precisely the same point. A chaotic attractor is no longer a simple geometrical object like a point, cycle, or torus and thus defies precise definition. The outstanding property of chaotic systems is that they are sensitively dependent on initial conditions: trajectories that emanate from two arbitrarily close starting positions diverge at a characteristic rate. Hence, chaotic behavior, although entirely deterministic, is inherently unpredictable.

Instabilities and Phase Transitions: Isolated Switches in Behavior

Phase transitions are accompanied by a colossal separation of time scales between different system components. Consider the case in which a certain macroscopic pattern is being replaced by another one. In dynamics terms, the first one becomes unstable and the second one becomes stable. Being close to an instability, however, implies that it takes a long time for a system to return to its steady state after a perturbation.3 The order parameters hence evolve arbitrarily slowly, whereas the underlying (real) subsystems maintain their individual, finite time scales. From the viewpoint of the order parameters, all the subsystems become arbitrarily quick so that they can adapt instantaneously to changes in the order parameters. The system dynamics thus amount to that of the order parameters, implying the ordered states can always be described by a very few variables if in the vicinity of behavioral changes. Put differently, the state of the originally high-dimensional system can be summarized by a few variables or even a single collective variable, the order parameter(s). The relationship between the subsystems and the macroscopic structure, in which the subsystems generate the macroscopic structure and the macroscopic structures enslaves the subsystems, implies a circular causality. This effectively allows for a low-dimensional description of the dynamical properties of the system of interest.

Complexity: Ongoing Switches in Behavior

The merger of dynamical systems with concepts of statistical physics has provided a thorough understanding of nonequilibrium phase transitions and hence of self-organized pattern formation. One can go a step further and investigate the case in which the system under study remains in a critical state, a phenomenon called self-organized criticality. Self-organized critical systems have a critical point as an attractor. Consequently, their macroscopic behavior resembles the temporal scale-invariance characteristic of the critical point, which often includes fractal dynamics (as in chaotic systems) and power laws for, for example, the system’s temporal correlations or spectral distribution (long-term correlations and 1/f-spectra).

Complex Networks

Macroscopic, scale-free behavior of complex systems may not only emerge as a result of self organized criticality. More recent studies on complex networks4 have revealed that certain forms of network growth yield scale-free networks; that is, the distribution of connections per node in the networks obeys a power law. The mechanism is that of preferential attachment: The likelihood that a new connection will formed with a node depends on the number of connections of this node. Thus, nodes that already have a large number of connections are more likely to get even more connections (“the rich getting richer”). Many real networks such as the World Wide Web, collaboration networks of scientists, and brain networks are probably scale-free.

Examples: From Physiological Rhythms to Match Play

Entrainment of Respiration and Locomotion: A Matter of Efficiency?

A scenario germane to dynamical systems theory is the spontaneous entrainment of respiration and locomotion cycles in mammals, also known as locomotion–respiration coupling (LRC). During walking or running, particular integer frequency ratios are adopted between cadence and respiration (e.g., 2:1, 3:1, 3:2, and so on). Recall the multifrequency attractors with commensurate frequencies explained above. Top athletes explicitly train their optimal frequency ratio and learn to flexibly switch between different ratios as a function of demand and strategy. Why would LRC be beneficial? For years, it has been hypothesized that it serves to enhance performance efficiency. More recently, this notion has been linked to optimization of the effective oxygen volume in the lungs. In brief, the cyclical abdominal pressure modulates self-sustained breathing and causes maximum oxygen concentrations at integer frequency ratios between cadence and respiration.

Over the years, it has become evident that LRC is not simply a result of (bio-)mechanical constraints, such as the vertical impulse arising from the footfalls, as entrainment of breathing has been reported for a wide variety of daily activities with markedly different mechanical constraints, including cycling, wheelchair propulsion, and rowing. In general, the degree of coupling and the resulting frequency ratios depend on a variety of factors associated with task, environment, and athlete. Theoretically, this implies that respiration– locomotion entrainment is a generic phenomenon that is instantiated by, but cannot be reduced to, specific mechanisms or processes. Practically, the finding that greater expertise is associated with a broader range of solutions renders this dynamic phenomenon intrinsically relevant to training and performance enhancement in all cyclical endurance sports.

Dimensionality Reduction: A Way to Understand Coordination and Perception

An expedient tool for studying complex patterns of coordinated behavior is principal component analysis (PCA), a statistical technique for identifying reduced dimensionality in multivariate time series. The dimensions or principal components that account for the most variance in the original data set may be interpreted as the degrees of freedom, or alternatively as the order parameters of the system under study. PCA has found wide application in the study of movement coordination, also in relation to sports, and appealing examples may be found throughout the literature.

PCA revealed how (bio-)mechanical constraints lead to a considerable reduction in the number of components required to describe human walking. Only four components proved to be sufficient for this purpose. Coordination patterns were found to change markedly from walking to running, and PCA revealed a profound reduction of dimensionality during speed-induced gait transitions, as discussed earlier. Importantly, the evidence for phase transitions was prominent in the coordination patterns but not in the stride parameters. This approach may be readily extended to gait patterns in competitive running, as well as other sports behaviors like catching, throwing, and hitting, as exemplified by several recent studies.

Quite a different application of PCA can be found in the study of visual anticipation skills in tennis, in particular shots in different directions (left, right) and to different distances (long, short). A few components appeared sufficient to capture most of the variance in the shots, but they differed across shots, and tennis players could predict shot direction based on these components alone. This suggests that visual anticipation skill in tennis, and most likely also in other sports, involves the extraction of low-dimensional dynamic information from high-dimensional displays.

This series of examples highlights the fact that (physical theories of) emerging patterns (order or disorder) can be utilized for decomposing complex movement patterns. Ultimately, a better understanding of the principles that govern the complementary processes—pattern formation and pattern decomposition—will help advance perceptual and motor skill training in sports.

Temporal and Spatial Correlations: From Individuals to Teams

Heart rate variability is a measure that is often used in sports practice to individualize training intensity and recovery. But how should it be understood form a dynamical systems perspective? Considering the complexity of the mechanisms regulating heart rate, it is reasonable to assume that its dynamics are nonlinear, although no firm evidence for this could be found. Instead, it appeared that the heart rate time series of athletes is characterized by long-term correlations with a fractal signature. In particular, scaling analysis (detrended fluctuation analysis or DFA) of heart rate time series revealed scale invariance in distinct regions, corresponding to well-known frequency bands in the power spectra of heart rate variability. Furthermore, during training, marked changes were found in the scaling exponents of the scale-invariant regions in question, as well as during recovery after heavy exercise. These findings suggest that (also) the scaling characteristics of heart rate variability might be used to monitor the training status of athletes.

Dynamics of Team Sports

Matches among sports teams represent highly complex dynamical events. The positions of the players on the field evolve over time and may be described in terms of their spatial, directional, and temporal characteristics. Although this is already a rather laborious exercise, it is not sufficient for an adequate understanding. The reason is that the actions of the players on the field only make sense in light of specific, yet not necessarily known, individual and collective objectives and strategies. In any situation, players have a multitude of solutions at their disposal to pursue their strategic objectives, while actions unfold in parallel. As a result, the dynamics of match play represent a notoriously difficult area of investigation. Nevertheless, although still in its infancy, the analysis of match play is on its rise due to both technological and conceptual advances. Notable among the latter is the advent of network theory, which concerns itself with the study of graphs as a representation of connections between discrete objects (nodes).

As a case in point, a network analysis of passes among the players of the Spanish team during the FIFA World Cup 2010 was performed, where the team was considered a network with players as nodes and passes as (directed) connections. This revealed the effectiveness of the Spanish game in terms of several network measures over time, most importantly the clustering coefficient and passing length and speed. Likewise, in searching for the greatest team in cricket, matches played from the late 19th century onward were analyzed as networks using a page–rank algorithm to assess the importance of the wins and the rank of teams and captains; the same approach is used, for example, by Google to rank websites by keywords.

Team sports do not only form a network as players interact during team play but also sociologically in that players of the same club are linked. The latter relations have been analyzed in soccer using bipartite networks with nodes being players and clubs. The probability that a player has worked at n clubs or played m games shows an exponential decay, whereas the probability that he has scored g goals represents a power law. If two players who have played simultaneously for the same club are connected by an edge, then a new network arises with an exponentially decaying degree distribution. Of course, performance (scoring) and social interaction (club sharing) influence each other.

Conclusion

The study of a system’s dynamics comes with a rich and sophisticated conceptual framework to address the complexity of sports behaviors. This is true for both the research questions that may be addressed and the methods used for that purpose. Absolute guidelines do not exist for this, only reasonable assumptions. Theoretical concepts may serve as sources of inspiration when linked to real-life phenomena and pertinent questions; conversely, the specifics of those phenomena and questions require an adequate conceptual and methodological approach. The focus on dynamical systems in sports science is still in its infancy but—given its marvels and intricacies—holds great promise for the future.

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