Quest Dynamic Systems Theory and Team Sport Coaching

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Dynamic Systems Theory and Team Sport Coaching a

Jean-Francis Gréhaigne & Paul Godbout a

b

GRIAPS, Université de Franche-Comté , Fort Griffon , France

b

Department of Physical Education , Université Laval , Québec , Canada Published online: 30 Jan 2014.

To cite this article: Jean-Francis Gréhaigne & Paul Godbout (2014) Dynamic Systems Theory and Team Sport Coaching, Quest, 66:1, 96-116 To link to this article: http://dx.doi.org/10.1080/00336297.2013.814577

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Quest, 66:96–116, 2014 Copyright © National Association for Kinesiology in Higher Education (NAKHE) ISSN: 0033-6297 print / 1543-2750 online DOI: 10.1080/00336297.2013.814577

Dynamic Systems Theory and Team Sport Coaching JEAN-FRANCIS GRÉHAIGNE GRIAPS, Université de Franche-Comté, Fort Griffon, France

PAUL GODBOUT

Downloaded by [Universite Laval] at 14:09 30 January 2014

Department of Physical Education, Université Laval, Québec, Canada This article examines the theory of dynamic systems and its use in the domains of the study and coaching of team sports. The two teams involved in a match are looked at as two interacting systems in movement, where opposition is paramount. A key element for the observation of game play is the notion of configuration of play and its ever-changing shape, namely through phases of contraction and expansion, and its moving location on the court or the field. Analysis of specific configurations of play by players, or external observers, is discussed in light of the notion of prototypical configuration of play and the process of learning by analogy. Offensive and defensive matrices of play, encompassing play in movement as a systemic whole, are presented as advance organizers that provide players with advance strategic representations of reference key points in the unfolding of game play. These various concepts, along with elements of movement in play, are integrated in a model intended to help players and observers grasp a systemic view of action play and its underlying fulcrums. Keywords Dynamic systems in team sports, prototypical configuration of play, matrix of play, team sport modeling

Over the past 20 years, there have been several publications discussing the use of the dynamic-systems theory for analyzing team sports. Gréhaigne (1989) discussed the potential contribution of systemic analysis to the study of game play in movement. This was followed by a further discussion on one particular aspect with the article “Dynamicsystem analysis of the opponent relationships in collective actions in soccer” (Gréhaigne, Bouthier, & David, 1997). Since then, the dynamic approach to systems in sports has been investigated by Tim McGarry and his collaborators (see McGarry, Anderson, Wallace, Hughes, & Franks, 2002) in response to specific questionings related to the analysis of game play (Franks & Goodman, 1986; Gréhaigne, 1988; Gréhaigne, Bouthier, & David, 1996; Hughes, 1996; Hughes & Franks, 1997). With respect to the theoretical basis of the analysis of game play, the use of concepts or constructs borrowed from other domains of research has not been truly examined epistemologically in order to clearly determine possible uses of this approach (Bourbousson & Sève, 2010). For instance, there should be some clarification concerning the relationship or analogy between “disturbances of a system” in a theory related to physics and mathematics and the analysis of an “opposition rapport” in game play. The question is: can a conceptual frame of reference conceived aside from physical activity and sports be used for creating specific and better adapted conceptual tools in the analysis and coaching of game play? Address correspondence to Jean-Francis Gréhaigne, GRIAPS, Université de Franche-Comté, 36 Rue de Tarsul, 21110 Izeure, France. E-mail: [email protected]

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Team-Sports Dynamics

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There are always problems with the simple application or import of theories put together in others fields of research. There must always be some epistemological consideration as to the scope and pertinence of concepts borrowed from an outside theory depending on the research or the professional application one wishes to conduct; otherwise, the use of such concepts remains a simple plastering for they frequently fit little or not at all with the field of research of interest. In other words, the matter of the rapport with concepts and knowledge cannot be properly analyzed if the matter of the operative value of such concepts and knowledge is not dealt with. With regard to team sports, the matter of the “rapport with knowledge” must therefore be examined at length and analyzed in light of differences, convergence and complementarities between the operative application of knowledge, which makes it possible to act in situation, and its predictive applications, which make it possible to put the players and the ball (or any other appropriate object) into movement while taking into account their properties, their relationships and the transformations they generate. The purpose of this article is three-fold: (a) to examine the theory of dynamic systems and its use in the domains of the study and the coaching of team sports; (b) to discuss, in relation with the perspective of “play in movement,” several conceptual tools that appear pertinent for the analysis of evolving and complex phenomena observed in team sports; and (c) to propose a coherent model of the dynamics of game play in team sports. The pertinence of using the theory of dynamic systems for the study and coaching of game play will be examined in connection with different concepts and analysis tools developed by what one might call “the Francophone school of team sports.” This school of thought rests on a pedagogical shift, in the 1970s–1980s, that consisted of considering that the opposition rapport was paramount in the study and teaching of team sports. That brought the need for better explaining play actions consciously undertaken by players in order to answer, during game play, to their opponents’ conscious and voluntary maneuvers. Throughout the article, the reader will note that the discussion is essentially playercentered. This in no way means that the coach’s involvement, or the coach’s orchestration as discussed by Jones, Bailey and Thompson (2013), is being ignored. Rather, it reflects that (a) the focus is on actual game play, or on play in movement, and (b) that the underlying pedagogical approach is constructivist.

Dynamics, Opposition, and Systems in Team Sports Whatever team sport may be concerned, an analysis of the dynamics of game play and of players’ choices sheds light on the rapport de forces1 (see Table 1 for a definition) and on the opposition rapports. Analyzing and explaining opposition, while considering it as a key element for any progress, represents a reference source for teachers or coaches: it makes it possible to conceive play situations, either at school or in training, that keep a learning focus while allowing the reality of opposition to be at play. Deleplace (1979) stated, we have had to demonstrate that it was necessary to meticulously analyze opposition rapports and that it was perfectly possible to make them explicit, to formulate and systematize them in view of a conscious and methodic use in training or initiation and during game play as well. (p. 9) Today, many experts agree more or less on the fact that to properly conduct analyses of team sports, one must have recourse to a dynamic approach of the confrontation in order to comprehend the organization of game play (see, for example, Araújo, Davids, & Hristovskic,

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J.-F. Gréhaigne and P. Godbout Table 1 Summary Table for a Definition of Unfamiliar Terms

Term Action sector

Configuration of play


Effective play-space (EP-S)

Intervention sector

Mother phase of play

Operative image Rapport de forces

Definition Also called “sector of play”; sector of the pitch or the court spatially defined by the limits of possible actions for the different attackers, within 1 s, when considering the player’s position, the direction of movement and the speed of movement. Schema drawn from the relative spatial location of players from both teams on the pitch and the location of the ball as well at an instant “t.” Polygonal area that one obtains by drawing a line that links all involved players located at the periphery of the play at a given instant. Sector of the pitch or the court spatially defined by the limits of possible actions for the different defenders, within 1 s, when considering the player’s position, the direction of movement and the speed of movement. Initial organization of game play; it is embedded in all others that will follow during a particular sequence of play. Functional mental representation that makes it possible to act on reality. Refers to the antagonist links existing between several players or groups of players confronted by virtue of certain rules of a game that determine a pattern of interaction.

2006; Davids, Araújo, & Shuttleworth, 2005; Frencken, Lemmink, Delleman, & Visscher, 2011; Glazier, 2010; Glazier & Robins, 2013; Kelso, 2000; and Travassos, Araújo, Vila, & McGarry, 2011). At the same time, it is recognized that much remains to be done, conceptand knowledge-wise, for the development of new analysis theoretical tools and renewed pedagogical and didactic maneuvers. Dynamic systems theory (see McGarry et al. [2002] for a review with regard to competitive sport, or Davids, Glazier, Araujo, & Bartlett [2003] concerning movement systems) usually designates a branch of mathematics that studies properties of dynamic systems. This active field of study develops at the frontiers of topology, analysis, geometry, and measurement and probability theory. The nature of the study is conditioned by the particular dynamic system concerned and depends on the selected tools (analytical, geometrical, or probabilistic). Walliser (1977) has named “système à état” (“state system”; p. 22) a classical system that evolves through time both • Determinist like, meaning that the state and the exit at an instant t depend only on entries (p. 23)


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• Causal like, reflecting the topical determinism of a system that makes it possible [for observers] to follow closely its evolution (balance, stability, and growth; p. 31). Its future (evolution) rests essentially on past and present phenomena. One must exclude here systems that are mostly random, displaying a discontinuous evolution through time. Team sports, with their self-organization and organization characteristics, and with players thinking or making subjective decisions that can alter the evolution of the system (Mouchet, 2003), related more to chaotic-like systems. Thus, two collective sport teams are not groups of particles functioning in a void. Clearly, being composed of a great number of players is not a sufficient condition for a system to necessarily tend toward balanced states. Nevertheless, it must be possible over time to follow the evolution of the state of a confrontation system since this system often converges toward a balanced state after a few oscillations. Players from these teams display an immensely complex behavior because they sometimes seem to wander at random and not so in other cases. However, they eventually reach a balanced state, in relation to a space called attractor, which simply characterizes a system reaching a stationary state. These attractors (a target, a point of equilibrium during static phases of game play, a space or a configuration of play [distribution of players on the field or the court] toward which a system evolves irreversibly in the absence of disturbance) characterize such systems that appear to follow at the same time deterministic laws and random laws. Such a condition makes difficult any long-term anticipation. Nevertheless, contrary to what has been stated by Davids et al. (2005), dynamic system theory, although very convincing from a physics point of view, cannot be considered as generally applicable and allowing the resolution of the problem of fundamental relations between dynamics and complex systems in team sports. In team sport, one is faced with a series of temporary stationary states as opposition rapports allow throughout the unfolding of game play. These states may reappear on a punctual basis. This is why we prefer to discuss the matter in terms of the dynamics brought about by the play in movement in a complex system rather than deductively applying, to game play, the theory of dynamic systems. Even more so if one considers that this so-called complex system will become less and less complex according to the information that one may obtain as to the state of the system and its probable evolution. Indeed, what is of interest rather relates to conditions that allow a description of the properties governing the evolution of the system towards balanced or unbalanced states; in fact, all things considered, the system displays a balance of a different nature depending on the kinetics of game play. Finite states of known configurations are necessary for players to act given the immense number of possible configurations of play (see Table 1). In this perspective, the expanding understanding of non-equilibrium states brought about much progress. If one assimilates order with balance, there is progress in shifting from order to disorder without completely destabilizing the team. One can also make a distinction between on the one hand the unbalance created in the opposition rapport with regards to an anterior configuration and, on the other, the dynamical balance recreated in the current offensive movement with an attempt to reorganize the offense to insure the continuity of game play. Such considerations show that the dynamics of play displays entirely new characteristics. Such a situation is well illustrated by the notion of “transition of phase” that similarly acquires a specific meaning only at the time of the shift of play with the activation of the double impact organization (Deleplace, 1966; Gréhaigne, 2009; Gréhaigne, Godbout, & Zerai, 2011). In game play, this rupture of symmetry relates to the notion of


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J.-F. Gréhaigne and P. Godbout

Figure 1. Dialectic aspects of game play brought about by the opposition (adapted from Gréhaigne & Godbout [2012]). Reproduced with permission from Dr Marie-Paule Poggi.

reversibility with regards to the previous phase of play but also the notion of a potential other reversibility in the coming phase. Finally, as illustrated in Figure 1, the dynamic confrontation inherent to team sports may be looked at from a dialectic point of view in many regards. Indeed, at whatever level of expertise the match is played, players’ tactical choices are constantly dictated by weighing up opposite dimensions of game play.

Play in Movement in a Complex System and Conceptual Tools for its Analysis Pushing further in the same line of thought, we can state that non-linear complex systems, or complex systems simply linear at times, may display unpredictable behaviors that can even look like random ones. Given that, one does not focus on the search for specific solutions but rather on finding answers to questions, such as “are there possible stable states,” “does the long term behavior of the system depend on the configuration of play”, or still, “does the mother phase of play (see Table 1) determine the evolution of game play?” In relation with this last point of view, an interesting and innovative aspect was recently developed by Frencken et al. (2011). It concerns the oscillations of the position of the center of gravity of game play in soccer and on the occupied area in small-sided games (concerning these subjects, one may also consult Gréhaigne, 1992). Analyzing play in movement in team sports, in a player-centered approach, requires tools that can be used both by the coach and by the players. Such tools should provide information relative to the unfolding of game play, taking into account the passage of time. This section of the article discusses several constructs that may help coaches interpret game play and make appropriate strategic and/or tactical decisions.


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Configurations of Play We define a momentary configuration of play as the schema drawn from the relative spatial location of players from both teams on the pitch and the location of the ball as well at an instant “t”. Moreover, as expressed by Gréhaigne, Richard and Griffin (2005),


if one considers a given configuration of play, one can summarize it using the notion of effective play-space for effectively occupied play-space by players . . . . The effective play-space (EP-S) (see Table 1) may be defined as the polygonal area that one obtains by drawing a line that links all involved players located at the periphery of the play at a given instant. (p. 62) By considering the respective positioning of attackers and defenders, it is also possible to determine an offensive effective play-space (OEP-S) and a defensive effective play-space (DEP-S; for more on this distinction, see Gréhaigne, Godbout, & Zerai, 2010). Considering configurations of play, one objective will be the description of more or less stable states of the opposition rapport. Those are configurations of play that, very temporarily, no longer evolve through time. One may think that some of these fixed points are attractors, which means that if the system approaches them, it will tend toward this state of balance. In the same way, we will consider periodic aspects, that is to say states of the system that repeat themselves after a while. Prototypical configurations of play are a particularly convincing example (Caty, Meunier, & Gréhaigne, 2007; Gréhaigne, Caty, & Marle, 2004) and will be discussed in a later section of the article. Concepts of opposition, continuity and reversibility are essential for the analyses of momentary configurations of play. Opposition relates to the confrontation in a dynamic system defined by a play area and time limitation in a match where two sub-systems confront each other with opposite interests and goals. Continuity of play and movement of play breed disorder, inducing disorganizations that tend to be compensated so long as defense can adapt. At times, however, what appears to be disorganized game play, especially on the part of the attacking team, may in fact hide some form of order that an experienced observer can make sense of. The purpose of each offensive action, for instance, is to induce and take advantage of an unbalance in the opponents’ defense system, to create a surprise effect and unpredictable situations in order to score. Attackers must make every effort to outrun the opponents’ reconstitution of a defensive balance or to place their defensive system in a critical position and thus upset the balance in their favor. Finally, with regard to reversibility, there is a loss of ball and passage to defense when attackers fail to regulate the perturbation they were trying to take advantage of. At that time, if immediate recovery of the ball is impossible, the team must reorganize itself in order to contain the counterattack with a good defensive falling back, in an attempt to reinstall a balanced state in the rapport de forces. Dynamic States in Configurations of Play The dynamic state concept (Gréhaigne, 2009) makes it possible to better understand how, at a given instant, players are moving. They occupy a location but this location evolves because at this instant t each of them displays an instantaneous and different speed of movement. Thus, the evolution of the dynamics of the system may then be modeled by conceiving a discontinuous evolution through time. It should also be noted that depending on the primary rules of the team sport concerned, the degrees of freedom of the system

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differ. The expression “degrees of freedom” refers in this case to the possibility for the system to evolve toward a direction partially forced upon by its spontaneous evolution. The analysis of the dynamics of game play makes it possible not only to perceive play regulation factors but also to apprehend various regulation levels and their interdependence. It equally leads to the replacing of the player’s or the group’s activity in a more global context that gives it meaning. Indeed, the adjustment to the reality of the instant is a process that develops at two levels at the same time:


• An internal process that is the organization or the reorganization of actions in order to reinstate the coherence of the player’s or the group’s responses with regards to the present and momentary situation • An external process that integrate these responses in a larger system represented by the confrontation between the two teams. In game play in movement, a seeming disorder often relates to a particular homogeneity other than the simple spatial distribution of the players on the pitch since it has to do with a distribution based on levels of speed. In opposition situations, speed rapports between players end up in having spatially non-homogeneous states compensated and stabilized. This means that these states would appear more homogeneous to an observer capable of decoding speed rapports whereas a classical observation would only show heterogeneous aspects structured at times by locations and geometrical shapes. With reference to and by analogy with concepts used by Planck (1941), such a distribution constitutes what we call a dynamic state or a “complexion” of play. One determines a complexion, or dynamical configuration, of play considering the distribution of the players and the ball on the play area with respect to their locations, their orientations, and their traveling speed within a given time interval. Complexions possess within themselves transformation possibilities that are at the same time limited by the evolution possibilities of the play at hand but important as well if one chooses a rupture by radically modifying the current movement. Of course, the circulation of the ball (or the puck) is of importance in team sport but as soon as the balance between the two teams sets up with a defense in block, the circulation of players takes precedence. In one second, the distance traveled by one or several players may be significant given their speed or their acceleration, as shown by spatial representations in Figures 2 and 3 in an ice hockey context. The modeling of play in movement, with notions of action sectors for attackers and intervention sectors for defenders (see Table 1; Gréhaigne & Bouthier, 1994; Gréhaigne et al., 1997), has made it possible to better understand the importance of the speed rapports between players in order to induce an evolution of the rapport of forces. With reference to Figures 2 and 3, it is obvious that in two seconds, with players on the move, the complexion of the play has changed a lot. Some players have moved little while others, moving at full speed, have contributed a considerable expansion to this dynamic state. One should note that the puck is passed in line with the receiver’s run in order to facilitate the continuity of the attack. Basically, movement gives weight to the attack, making it possible to pass through the defensive block either through strong pressure on a weak link or through acceleration. The characteristic of the players’ circulation is that it can be at the same time very fast but interrupted with a series of stops. Then, transitions of play between two states (two momentary configurations of play) become essential observation elements for appropriate decisions. Block/pursuit, advanced/delayed position, stop/movement become key observation elements.



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Figure 2. A configuration of play at T0 (color figure available online).

Contractions and Expansions of Game Play Another aspect of game play seems interesting to explore from a complex system point of view: it relates to transitions. Transitions and transition play often relate to configurations in which one has some time at one’s disposal to take action due to a lower density of players. Attackers must then take advantage of the moment of unbalance to maintain their potential advance as defenders must rapidly come back, or stay, in block. Resorting to long- and/or short-play rapidly transforms configurations. As soon as long-play becomes possible, configuration shapes evolve and, most often, exhibit an “extended” play situation. For its part, the “contracted” model of play is often seen when game play occurs in a stabilized area with a significant density of players. Often, in this type of play in movement (contraction/expansion), play actions succeed one to another. For instance, in rugby, it is classically designed as grouped/de-grouped, representing either a concentration or a dispersion of the players. The notion of density


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Figure 3. A “complexion” of play at T+2s (color figure available online).

would relate rather to a qualitative aspect of game play for it closely depends on the quality of the players involved in an action in a given environment. For their part, notions of concentration/dispersion seem rather related to a quantitative approach in terms of number of players involved in the current action in a specific area, as illustrated in Figure 4 in an ice hockey context. Thus, one would associate contraction/expansion to movement and concentration/dispersion to the series of static states (or to discontinuous temporal aspects). These situations seem to function in close symbiosis and may help decipher and anticipate movement in game play in order to make judicious decisions.

Prototypical Configurations of Play and Learning through Analogy Time at one’s disposal for school teaching or coaching in team sports is relatively limited. Players, therefore, must rapidly construct knowledge and motor competencies. Moreover,



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Figure 4. A series of contractions/expansions in ice hockey (color figure available online).

once constructed, this knowledge must be used as a reference base for other faster learning with regard to the management of momentary configurations of play. This is important since it is likely to optimize players’ activity, given the dynamics of play in movement, allowing them to characterize momentary states of the rapport de forces and their probable evolutions. The methodology used to study configurations of play involved the reviewing of video recordings and the selection and analysis of play sequences that lead to a shot on goal or a goal in small-sided games. From the studied sequences of play, it was possible to identify configurations of play that appear most often with “coping players” (i.e., players that, although not experts, are no longer beginners either). These configurations that appear periodically and are essentially temporarily stable states have been labeled prototypical. Each represents an original model, archetype of a model that periodically reoccurs (Gréhaigne, 2007; Gréhaigne, Caty, & Godbout, 2010; Gréhaigne, Zerai, & Caty, 2009; Moniotte, Nadeau, & Fortier, 2011; Zerai, 2011).


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Knowing these configurations makes it possible for players to construct prototypes through the categorization of geometrical shapes, of classes of object properties and finally of temporal relationships in view of becoming more efficient during game play. These configurations are momentary and as game play unfolds, they link one to another. Nevertheless, for one to decipher them—or as Jones et al. (2013) would express it, “notice” them—one needs reference marks, operative images (see Table 1) that will facilitate the decoding of the perceived situation in view of a fast response. A few signs make it possible to rapidly identify a situation of play and its potential development. Thus, it is possible to draw the players’ attention to key points that may prove useful before the action in order to understand, organize, prepare, and respond rapidly to the requirement of the play or of a particular movement. Consequently, play models may be visualized by manipulating images likely to make students or players reflect on how they should proceed—to bring them to understanding in order to succeed. To this effect, over the past 30 years, the notion of analogy has become an important point of interest and has resulted in many studies in relation with learning (e.g., Gentner, Loewenstein, & Thompson, 2003; Gross & Greene, 2007; Kurtz, Miao, & Gentner, 2001; Liao & Masters, 2001), with cognitive science (e.g., Day & Goldstone, 2011; Gentner, 1983; Gentner, Holyoak, & Kokinov, 2001), with neuroscience (e.g., Bunge, Wendelken, Badre, & Wagner, 2005; Green, Fugelsang, Kraemer, Shamosh, & Dunbar, 2006; Speed, 2010), and with artificial intelligence (e.g., Hinrichs & Forbus, 2011). It is a matter for the learner to draw new conclusions based on similarities between two things. This process takes place in everyday thinking, learning, and problem solving. Once the link has been established, a new response may be elaborated from one or several responses already constructed. In situations where one has some time at one’s disposal, or for putting together strategies (in pre-planning before the action takes place), it is also possible to make use of procedures related to meta-recognition (Cohen, Freeman, & Wolf, 1996). The recognition/meta-recognition model deals with time-stressed decision making. This decision making would be a problem-solving process that starts with the recognition of configurations of play used to coordinate in advance the players’ actions (strategy making) and search for additional information, if necessary. These strategies, planned before actual game play, help identify problems of conflicts or of dependability with regard to the coordination of collective actions in the situation of play at hand, and may lead to the adjustment of the responses considered. In players, the end result is the grasping of a series of configuration models and especially the understanding of the strengths and weaknesses of each of them. More precisely concerning the players’ thinking, they can choose a two-level strategy: (a) either the activation of the cognitive meta-recognition with held over associated responses or (b) an optional process of critical analysis and correction. Together, these processes construct, verify, and modify play “scenarios.” This way, a player may face a series of relatively new events. The study of a complex system dynamics in team sports requires tools for predicting with accuracy the evolution of the rapport de forces. In this respect, an appropriate mental functional (i.e., constructed in view of action) representation of configurations of play and their potential evolution is critical for players’ efficiency. Ochanine, Quaas, and Zaltzman (1972) have discussed the notion of functional representation in terms of an operative image that comprises three main characteristics (finalization, laconism, and functional distortions): • Finalization is the main property of the operative image. In fact, Ochanine et al. distinguish the cognitive image from the operative one; whereas the first is the integral reflection of the object, the operative image is constructed during the action on


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that given object. Finalizing the operative image results in the selection of pertinent information. • The operative image retaining only what is directly useful for action, it is thus selective. Compared to the cognitive image, it is laconic and task centered. • Functional distortion refers to the accentuation of the most important informative elements with regards to the task at hand: properties of the object, its different aspects, and its partial structures. Its purpose is to remove or reduce to a minimum the uncertainty of the action situation. The model put forward by Ochanine et al. appears to be quite complementary to the one presented by Cohen et al. (1996). It distinguishes two types of operative images depending on their function in the treatment of information during the action: afferent images and effector images. Afferent operative images govern, or condition, successive states of the configurations of play, whereas effector operative images govern the selection and preparation of action decisions. From this point of view, offensive and defensive action matrices, that provide the synthesis of elements of game play as a whole, are a must to help recognize and solve problems brought about by momentary configurations of play. Matrices of Play as Advance Organizers Deleplace (1979, 1994) has proposed, as a first form of representation of game play, the notion of system of matrices. Matrices (in the genetic sense of the term, that is, the source of knowledge) “constitute a coherent system of the mental representation of the whole logic of game play” (1979, p. 21). Matrices represent a true systematic of tactical decisions in play. A matrix of action is useful for embracing the entire complexity of game play in a strong functional logical unit. Such a system must make it possible to redeploy, whenever needed, the understanding of game play in full action, considering any of its fundamental axes. In this context, the defensive matrix is the collective organization of the most simple and, at the same time, the most general defense in order to impede, whatever its shape, its deployment or its successive rebounds, the offensive movement attempted by the opponents momentarily in possession of the ball or the puck. Inversely, the offensive matrix is primarily a choice as to a way of penetrating into the adverse defensive system given its momentary configuration. To the principle of moving about between the defensive lines while staying in block, in the defensive matrix, corresponds, in the offensive matrix, a principle of transformation of movements in order to overtake the defenders’ replacement. From this point of view, offensive and defensive matrices are based on dynamical principles of organization during game play (Figure 5). In play, the covering of the pitch in defense as well as the occupation of this same area in the attack focus on two dimensions to be taken care of although partially contradictory: the width and the length in relation with the dispersion and/or the concentration of the players of each team. In soccer, for instance, this means taking care constantly of the width and worrying more periodically about the length depending on the organization of play. Over the past decades, the fields of educational psychology and educational research have explored the notion of “advance organizers” originally developed by Ausubel (1960). An advance organizer is a more or less general model that helps students deal with information by linking it to a larger cognitive picture that reflects the organization of a given subject matter. By analogy, an advance organizer in team sports would be a frame of reference that helps players organize perceived information in view of responding more efficiently to


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Figure 5. Game play and matrix system in team sports (adapted from Gréhaigne & Godbout [2012]). Reproduced with permission from Dr. Marie-Paule Poggi.

problems brought about during game play. Both offensive and defensive matrices may be thought of as advance organizers with respect to team sports in the sense that they provide, for the players’ benefit, a pre-existing frame of reference to which they can refer when time comes to organize their response to game play. To a certain extent, both types of matrices may be thought of as a general pre-strategy on which a more specific strategy and punctual tactical decisions may be based (Gréhaigne, Godbout, & Bouthier, 1999).

Organization and Self-organization of Game Play Contrary to spontaneous and free play among children, team-sport game play requires organization if one wishes to avoid chaos. Efficient decision making in terms of strategy and/or tactics requires self-organization on the part of the players, beyond what Jones et al. (2013) called coaches’ orchestration. However, players’ decisions are neither taken at random or on a totally free-will basis. With regard to the organization of the players’ actions in a given team sport, rules have been established in order to provide a structure that will facilitate game play and guide players’ actions. Called primary or fundamental rules, they constitute the foundations of play organization. These constraints limit, guide and regulate players’ actions; they are delineated in a restrictive rather than prescriptive way, allowing players to experiment with varied responses. For instance, in soccer, “one cannot touch the ball with one’s hands (except for the goal keeper and the player responsible for the throw-in)”, rather than “one must touch the ball with one’s foot”. Once they know what is forbidden, players can get creative and vary their actions while abiding by the primary rules. Primary rules concern: • Scoring, in relation with specificities of the targets of the team sport concerned (these targets bestow their characters to the opposition rapports). • Attackers’ and defenders’ movements that proceed from the logic of scoring and complete it, in acknowledgment of the necessary equality of chances. • Degree of action freedom on the ball, that makes it possible for game play to be alive in a specific way by promoting or not the continuity of movements. • Terms of physical engagements, that insure the respect of the “athletic bias.”



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For their part, secondary rules or conventions facilitate the normalization and the evolution of the unfolding of the sport. They differ from constraints in that they may be modified without putting fundamentally in jeopardy the essence of the game. Two main categories of secondary rules are: (a) action rules and their related principles of action (e.g., reducing the number of exchanges required to reach the scoring zone, as part of “playing in movement”) and (b) play organization rules (e.g., assigning an optimal position on the court for each player). For a more elaborate presentation of these rules, one may consult Gréhaigne and Godbout (1995). Concerning players’ self-organization of play and from a temporal point of view, most common phenomena are either the emergence of multiple stationary states, or the evolution towards changes corresponding to a stationary state that has become unstable following some disturbance in the evolution of game play. These are known phenomena and happen at all levels of the organization of play. All phenomena based on variations of rhythm may be interpreted as the emergence of a temporal order, in the form of continued oscillations. Amazingly, one can see that whatever the nature of the elements of the play system considered non-linear interactions between elements often lead to identical phenomena. The reason for having so recurrent periodic phenomena in team sport seems thus related to the non-linearity of the regulation processes. Since non-linearity gives rise to sometimes multiple solutions, one must expect varied transitions between units of play as well as the coexistence of qualitatively different tempos. Because of these phenomena, nonlinear systems are capable of displaying a complex behavior that often translates not only into diversified configurations of play but into configurations that regularly recur as well. The problem then, for players, is to decode these configurations and, whenever possible, associate them with known prototypes. The situation is much different when a system (or a team) functions far from a balanced state. Naturally, the system will attempt to react to this instability by evolving towards a balanced state. However, throughout this evolution where disorder prevails, the team may display an unexpected behavior during a more or less extended interval of time. For instance, the attack may maintain the advance taken on the opponents and thus the system will fail to reach balance as long as constraints apply and attackers can move towards the scoring zone. Inversely, defenders may also reestablish balance by coming back into block defense or by recuperating the ball. So, it appears clearly that in the organization of play, a deviation from balance may lead to new possibilities of complex behavior(s) through a rupture of symmetry. More specifically, one can recognize the overtaking of the critical threshold for the deviation from balance with a recuperation of the ball in a promising location of the pitch, a crossing of the “advantage line” or the fast placement of the ball in front of the effective play-space. A particular phenomenon that may lead to the creation or maintenance of unbalanced states in game play has received a good deal of attention from sport psychologists; it relates to the notion of psychological momentum (PM; see, for instance, Gernigon, Briki, & Eykens [2010] and Moesch & Apitzsch [2012] for excellent reviews). Loosely defined by Adler (1981) as “a state of dynamic intensity marked by an elevated or depressed rate in motion, grace, and success” (p. 29), PM has also been defined by Iso-Haola & Mobily (1980) “as an added or gained psychological power which changes interpersonal perceptions and influences an individual’s mental and physical performance” (p. 391). Summarizing their review of the literature about research on psychological momentum, Moesch & Apitzsch (2012) wrote:

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There is both empirical evidence for the existence of PM . . . and its relationship to performance . . . as there is against those predictions. However, already Adler (1981) in his basic work on Momentum stated that it is not significant for the development of momentum if it really occurs, but much more if it is perceived to occur by the subject. (p. 437) For their part, Gernignon et al. (2010) have examined the dynamics of PM in sports from the dynamical systems theory perspective. They wrote: “In spite of the apparent similarities between properties of PM and those of dynamical systems, PM cannot be viewed as a dynamical phenomenon in perfect accordance with the theory of dynamical systems” (p. 380), a position similar to the one we have expressed earlier with reference to the analysis of the dynamics of team sports. Finally, whether a PM effect may be objectively established or not is, to a certain point, irrelevant in this paper in the sense that if it is indeed present, it may be one of many possible factors causing a system (or a team) to function far from a balanced state. In play actions, as is the case with several phenomena where non-linear physics laws apply, disturbances occur from the moment the speed of play reaches a certain point. There seems to be a critical threshold up to which the speed of play is bearable and all of a sudden, adaptation disappears. In a word, game play seems to enter in a state of instability. One may see this happen either at the match system level (the confrontation between the two teams), at team level (organization within the team), or at a partial forefront level (confrontation between some players at the forefront; for a discussion on these levels of organization in team sports, see Gréhaigne & Godbout [1995], Gréhaigne & Godbout [2013], or Gréhaigne, Godbout, & Bouthier [1999]). Thus, one can observe inside a phase of play spectacular phenomena leading for a few seconds to a particular temporal order following disturbance actions (long pass, crossing the ball back from the goal line, and forward-ball play) contrary to usual criteria of stability. Hence, in the (presumed) absence of rupture between the phases of play, one observes a system behavior characterized by a uniform distribution of the various properties of the system throughout space and over time. Should one voluntarily create a disturbance in order to induce moving away from this balanced state, the system always develops mechanisms intended to damp down the disturbance. In summary, the conception of time and time-related notions take up new and original characteristics that shed light on the value of dynamics in the study of game play. Given what has been written so far in this article, notions of speed, variations of pace (acceleration, deceleration), and disturbance become critical contents for the didactics of team sports. In play in movement, the speed of the ball and shapes of its trajectory constitute also unavoidable temporal aspects of good learning. Teaching movement-centered game play requires that one question one’s teaching or coaching “traditional” habits (Bourdieu, 1972).

Team Sports, Dynamics, and Analysis of Game Play: A Model In team sports, collection of information through observation, either by the coach, by fellow players or by players directly involved into action play (Gréhaigne & Godbout, 1998) is essential for appropriate decision making (Gréhaigne, Godbout, & Bouthier, 2001) either in a context of strategy planning or in one of tactical decision (Gréhaigne et al., 1999). Interpreting play in movement requires a systemic view of action play such as one illustrated in the following model (see Figure 6). In the central part of the figure, only one



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Figure 6. A model for the observation and analysis of movement in play in team sports.

configuration of play, with its respective offensive effective play-space and defensive effective play-space, is illustrated for clarity. During the sequence of play, this configuration normally changes constantly depending on the contraction or expansion of the effective play-space and its displacement backward or forward on the field. From a practical point of view, assuming a vocabulary centered on dynamics implies grasping correctly the terms used to discuss placements and displacements in game play, given the conditions of the confrontation. Movement in play creates dynamics according to the distribution of the players on the field and to the speed rapports among them (Gréhaigne et al., 1997). One way of characterizing action play consists in defining the various microstates of the attack/defense system with, for each micro-state, the analysis of the players’ distribution on the field according to three parameters: their position, their orientation, and their speed of displacement. This refers to a temporarily stabilized dynamic configuration that makes it possible to anticipate the evolution of the rapport de forces. At a good level of play, an apparent disorder, in the physical sense of the term, represents then a particular homogeneity other than a simple spatial distribution or regrouping of players on the field. One then refers to a distribution with respect to energy levels based on the construction, by players, of a temporarily stabilized transition. In opposition situations, these fundamentally energetic interactions are related to players’ speeds and accelerations/decelerations. This means that these displacements would appear homogeneous to an observer capable of decoding energy levels (speed volumes in interaction) whereas a classical observation


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would only show details heterogeneous but structured according to positions and geometric shapes. Here, one is concerned with dynamic states and, essentially, mobility and displacements are the elements that insure balance between defense and attack. A series of momentary configurations of play between gaining possession of the ball and its loss represents a sequence of play. In the analysis of a sequence of play, reducing the decision cascade to a binary choice or to a simple study of player dyads weakens the analysis or appears too formal with respect to all possible actions in team sports. This would mean making the hypothesis that game play, according to players’ actions, always tends towards the most probable configuration. One should then deduce that the entropy (disorder) related to the configuration of the system is stable. Now, the variation of the entropy of a system is related to the probability of transition from one momentary configuration to another one also related to probabilities of the appearance of other configurations. In particular, as sustained by Deleplace (1979), the more the transition probabilities between two configurations strongly differ depending on the direction of the transformation, the less probable the transformation will appear. However, the appearance of an unlikely configuration may be facilitated: • By the existence of a sub-set of configurations, from likely configurations to unlikely ones. • By the duration of the confrontation: the longer it lasts in a given sequence of play, the less predictable a specific situation will be. • By an unexpected event (e.g., a counterattack) that disrupts some stability of the play, meaning it increases disorder and facilitates the appearance of an unexpected configuration. Thus, willingness to reduce disorder in game play at the advantage of some external order or some prescriptive model must not lead to the neglect of entropy in the modeling of game play.

Conclusion The successive distortions of the effective play space and the evolution of the configurations of play, as well as the elasticity inherent to game-play shapes, constitute privileged indicators of the confrontation. Players are no longer conceived as related to a fixed space but rather as moving bodies occupying the play space in a setting of potentialities (Fernandez, 2002). Thus, the purpose of this article was to discuss the dynamic systems theory and its direct application in the domains of the study and the coaching of team sports. A second purpose was also to synthesize and/or create more pertinent conceptual tools to analyze complex evolving phenomena. In team sports, based on some known framework of play, according to the importance of the encounter, to the evolution of the score, to the passage of time and to their perception of the rapport de forces, players can, up to a certain point, change or adapt the agreed upon strategy by displaying such and such behavior that reflect different tactical choices. Here, the model considered in this paper appears critical for deciding fast and correctly. One facet of these dynamics implies that the behavior of the systems converges toward states that tend to display, at times, some stability over time. In other respects, a match is by definition open and subject to external influences. Its dynamics is also dependent on fluctuations closely and necessarily related to the functioning of sub-systems (the


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match, the partial confrontation or, yet, series of one-on-one duals). Without such analyses, researchers, coaches, and even players would not dispose of significant information. Finally, it should be obvious that given this dynamic view of team sports, coaching/learning setups should make provision for dynamic situations where temporal pressure is a major component.

Notes 1. In previous publications, we translated this concept as “rapport of strength” or “force ratio.” After long discussions with Anglophone colleagues, however, it seems better to keep the French term and explain what it means.


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