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  clubs  or  played  games  shows an exponential decay, whereas the probability that he  has  scored  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.


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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