Abordagem de sistemas complexos lineares e nao lineares para o esporte - Educação Física (2024)

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/240589732Linear and nonlinear complex systems approach to sports. Explanatorydifferences and applicationsArticle · January 2012CITATIONS2READS1,0806 authors, including:Some of the authors of this publication are also working on these related projects:effects of the processed sugar View projectIntegrated training View projectRobert HristovskiSs. Cyril and Methodius University114 PUBLICATIONS2,441 CITATIONSSEE PROFILENatàlia BalaguéNational Institute of Physical Education (INEFC)129 PUBLICATIONS791 CITATIONSSEE PROFILEVujica ZivkovicSs. 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It is needless to say thatbiological and socio-biological systems in sportlike athletes and teams belong to this class.However, components may interact in two generalways: linearly or nonlinearly. The linear approachto complex biological systems started with theadvent of old cybernetics (e.g. Ashby, 1966) with-in the research program headed for finding com-monalities between the living organisms andmachines. This approach today is mostly repre-sented by the theoretical and engineering, i.e.applicative, work within the realm of the controltheory. The basic premises of this field of researchis that living organisms may be successfully under-stood and modeled as technical devices decom-posed on control and controlled components withexplicit feed-forward and feedback loops amongthem. Components themselves are usually present-ed in a form of spatially encapsulated, pre-formedentities which activate or deactivate as soon astheir function is being affected by other clearlydefined subsystems. The analogy of anatomical-structural and functional identity is tempting andthus represents a fundamental assumption withinthis approach. A strong pre-formed structure-func-tional compartmentalization and modularity ishence hypothesized. In other words, each structureis a pre-formed module that serves one and onlyone function and is wholly responsible for its func-tional performance. Thus, its contribution to theoverall performance can be isolated and linearlycombined with other component contributions giv-ing a linear superposition, i.e. a sum which thendefines the overall performance. This is also theassumption behind all statistical methods of theGauss-Rao linear model. Linear systems are pro-portional in a sense that the output of the system isalways proportional to the input. If the input iscontinuous then the output is also continuous andvice versa, i.e. for a small or large change to bedetected at the output a small or large change in theinput is required, respectively. Small change at theinput can not produce large, not to speak a qualita-tive, change at the output. In sports realm thismeans that the performance should be proportion-al to its causes (inputs).For the components giving rise to the system’sperformance to be successfully controlled, a super-ordinate control device must contain the wholespecific knowledge in an explicit, rule-based way.It is important to understand that the program, bydefinition, contains detailed specific instructionsof how each component should behave. Hence, byusing feed-forward specific commands/programsit would affect the state of controlledcomponentsand eventually the overall behavior of the system.Hristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 25LINEAR AND NONLINEAR COMPLEX SYSTEMS APPROACH TOSPORTS. EXPLANATORY DIFFERENCES AND APPLICATIONSUDC:796.011.1:37.013(Review)Robert Hristovski1, Natalia Balague2, Boro Daskalovski1 Vujica Zivkovic1, LenceAleksovska-Velickovska1, & Milan Naumovski11 University of “Ss Cyril and Methodius” - Faculty of Physical Education - Skopje2 Institut Nacional d’Educatio Fisica de Catalunya, Barcelona, SpainAbstractWe present a brief pedagogical overview of the major conceptual and explanatory differences of linearand nonlinear dynamic complex systems approaches to sport and define some applications which arealready under development within the nonlinear approach. Key words: sports, linear and nonlinear dynamical systems, complex systemsThese components then would feed back the infor-mation about their current state or change of thestate, so that the control unit detects errors andfeeds forward new correcting commands/programsdownwards to them. That there are efferent andafferent pathways in biological systems is a wellestablished fact. However, the problem arises inunderstanding the dynamics of such systems andthe functions that are formed as a result of it. Thepre-formed hierarchy is a necessary ingredient insuch linear complex dynamical systems with anencapsulated super-ordinate controlling unit, i.e. acentral governor, on its top. Needless to say, sucha structure is subject to fundamental explanatoryproblems, such as the infinite regress or the“homunculus” problem: If the super-ordinate con-trol device programs the subordinate devices, i.e.system’s components, then who programs thesuper-ordinate device itself? Who puts the specificknowledge into it? For technical systems theanswer is easy: computers are programmed byhuman beings called programmers, and computerscan control a whole factory. In this sense program-mers are super-ordinate to computers. They put theknowledge into computers. But who programs liv-ing systems? Two answers may be offered. One isthat another, higher controller, is responsible forprogramming them (e.g. the brain). However, thisexplanation just shifts the same problem to a high-er level and remains unresolved. If this is true, thenwho programs the brain? Another answer wouldbe: The super-ordinate device self-programs, orbetter to say, self-organizes. Here, it is very impor-tant to note that this super-ordinate device does nothave to be pre-formed at all, although it would nec-essarily depend on extant biological material sub-strates like the neural, metabolic or hormonal sys-tems and their components. The super-ordinatedevice may self-organize or emerge under somesuitable conditions. Also, depending on the condi-tions, different super-ordinate devices may form,stabilize and dissolve from the same componentsof the system. This possibility and evidence for itwill be discussed more extensively in the next sub-heading. The control theory also deals with possiblenonlinearities by explicitly defining their type forone or few system’s components (see Agrachev etal, 2004). However, this is an ad hoc solution thatmerely mimics the nonlinear behavior. In engineer-ing, this mimicking is acceptable solution due tothe tasks and problems under scrutiny. One has justto project and construct a technical device that haspredictable and controllable behavior. Thus, bydesigning such mimicking device with inherentnonlinear behavior one solves the problem.However, the task of explaining the behavior ofliving systems such as athletes or sport teams isessentially different. One has to explain how suchbehavior emerges in the first place, without ad hocinterventions. In other words, the task is not one ofconstructing but one of explaining a certain extantliving system. The researcher has to construct anomothetic system, i.e. a theory, which hasexplanatory as well as predicitive power that cap-tures such phenomena in a general way. Not mere-ly finding special, ad hoc, explanations for eachseparate phenomenon, but capturing those separatephenomena as special cases, i.e. instantiations, ofa general explanation. Of course, the linear approach is obviously suit-able for constructing engineered systems (telecom-munication systems, computers, automated controlin factories etc.) where an external explicit design-er or programmer in a form of a human being isengaged, but the inverse assumption, namely, thatliving organisms can be approximated as engi-neered machines does not need to be necessarilytrue. A support for this claim comes from decadesold and unsuccessful endeavor of roboticists andartificial intelligence researchers to construct arealistic biological isomorph, i.e. a system whichwould behave as a real biological system. Thisproblem is put to an extreme in the area of roboticmotorics where biped robots are announced as ‘thestate of the art’ if they are able to perform skills ofa one year aged toddler. The claim of the pioneer-ing researchers in these fields that problems wouldbe solved once the processing and computationalabilities of artificial systems become powerfulenough seems a far fetched assumption.Supercomputers today outperform their earlyancestors by a factor of million, and yet no biolog-ical isomorphs are in prospect. Maybe the coreassumption of the technical-biological systemanalogy is a wrong one. Maybe living systems arenot technical systems after all. In robotics engi-neering this was understood as a major challengeand alternative, biologically inspired, ways of hier-archical self-organization started to take place inthe 1990-es and more recently (e.g. Paine & Tani,2005). Nevertheless, the linear approach to livingHristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 26complex systems have had a wide impact over thepast several decades in different areas of biologyand its subfields and consequently in sports sci-ences in general. However, the new wave of bio-logically inspired engineering have already starteda new era of building technical systems. The epi-logue, so far, may be concisely formulated as:Instead of explaining and modeling biological sys-tems as technical systems lets try to engineer tech-nical systems as real evolving biological systems.Time will tell whether this endeavor is feasibleand, if yes, to what extent.Without disregarding the insights produced bythe experimental work in a vast set of scientificdisciplines investigating biological order withinlinear control theory approach, it seems increas-ingly likely that it is the dynamics of these systemsrather than their structure that gives the essentialinformation on the real basis of their functioning.Of course, the structure, being dynamic itself and aproduct of that very dynamics, may be a strong,even sometimes the dominant constraint on thebiological dynamic order (Edelman, 1987,Edelman et al, 1999). However, it seems, that with-out finding dynamical laws that create, maintainand dissolve the biological functional order, onecould not proceed further even to explain how bio-logical structures themselves come about in thefirst place.The complex nonlinear dynamic systemsapproach to biological systemsIn the previous text we defined the problem offinding dynamical laws that create, maintain anddissolve the biological functional order. More pre-cisely, we asked: what are the dynamical laws ofthe creation, stability and dissolution of biologicalorder? Or, even more precisely, what are thedynamic laws responsible for these phenomena(e.g. performance) in sports settings? First andforemost, is their dynamics linear or not? Letsmake it simple: If one findsnon-proportionaleffects in systems under scrutiny their dynamics isnon-linear1. The answer comes quickly: Sportsitself contains innumerable non-proportionate phe-nomena, such as: action selections i.e. decisions,sudden task disengagements under accumulatedeffort, onset of overtraining, stable and unstableperformance profiles, emergence of a new tech-nique or a movement form, lactate and ventilatorythresholds (thresholds are hallmarks of nonlineardynamics) to name a few. All these phenomenaarise when a small change of certain parametershave taken place. For example, a small change inthe distance from the opponent may bring about aqualitative reorganization of the action. A smallchange of the exercise intensity brings about theonset of blood lactate accumulation. These factsmake sports a fertile ground of investigation ofcomplex biological nonlinear dynamical systems. What are the characteristics of nonlinear bio-logical systems? First, interactions between theircomponents are non-linear. Components in suchsystems influence other components, but in returnare being influenced by them. Hence, componentsposses a property of self-influence, sometimesdirect, but generally indirect. The number of suchcomponents is vast. Only the brain contains hun-dred of billion neurons and thousands of timesmore synapses. Making an explicit and detaileddynamic model of such complex system is unfea-sible in foreseeable future. Furthermore, the brain-body system is embedded in an environment whichis infinitely variable in space and time. To adapt tosuch environment a linear system with pre-formedrule-based devices would need to contain infinitenumber of programs or rule-based representationsto perform the adaptive flexibility required by thetask and the environment. However, nonlinear sys-tems have one general characteristic which enablessuch flexibility. It is called multistability. This termsignifies the ability of nonlinear systems to havemultiple semi-stable states for the same or similarvalues of some contextual variable that representsthe environment. This means that such systemsmay perform more than one function in a same orsimilar environmental context. For example, onecan perform different movement organizations tosatisfy the same task requirements. Moreover, one can switch between such statesand show significant behavioral versatility i.e.demonstrate multifunctionality. This is enabled bya dynamic mechanism called a meta-stability(Fingelkurts & Fingelkurts, 2004, Freeman, &Holmes, 2005, Friston, 1997, Izhikevich et al,2004, Kello et al, 2008, Hristovski et al, 2009). Forexample, one and the same neural network mayproduce a large number of patterns depending onthe inputs i.e. external constraints. The multistablesystem can dynamically shift among availablecomponent configurations which are selected bythe confluence of constraints that impinge onHristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 27them. Note that in this case there is no need of apre-formed super-ordinate device, a knowledge-able homunculus, which will make the selection. Itis the immediate constraints that make a selectivepressure and mold the system’s configurations tobe functional. The notion of constraints is instru-mental for understanding the behavior of nonlinearsystems such as athletes and teams. It is their roleto form the context within which complex nonlin-ear systems such as athletes, self-organize in a cer-tain functional way. Constraints may be classifiedas: task and environmental demands and orga-nizmic properties (i.e. athlete’s morphology, psy-chological traits, motor abilities, synaptic net-works etc.) (see Newell, 1986, Chow et al, 2011).Their interaction creates the web of influenceswhich form the control parameter space of the sys-tem. Now, if we change these control parameters ina certain way we can witness interesting phenom-ena. For certain values of control parameter, thesystem, i.e. athlete or a team, changes slightly ordoes not change at all (another example of non-proportionality). Then, for a further minor changeof the control parameter a qualitative change of thebehavior occurs. This is called a critical point. However, here we are challenged to define thebehavioral variable or variables we would use tocapture system’s behavior. In complex biologicalsystems there is a vast set of potential variables. Ithappens that the variable which shows an abruptchange is the relevant, essential, variable that bestcaptures the system’s behavior. This claim has arigorous mathematical justification. It is called a“central manifold theorem” and its physical, chem-ical, biological and sociological interpretation isarticulated in Haken’s “enslaving principle” (seee.g. Haken, 2000). This principle says that: close tocritical points only one or a few variables enslaveall other variables (system’s components ordegrees of freedom) and make them behave coop-eratively, so that to enable their own existence. Inother words, a large number of componentsbecome dependent on the behavior of only one orfew components. Then, by knowing the behaviorof these few components we actually have infor-mation on the behavior of all other components.They behave cooperatively as the enslaving, col-lective, variable dictates. What happens is a spon-taneous separation within the system on vari-able(s) that govern and variables which are beinggoverned. Variables which govern are called orderparameters because they reflect the ordered behav-ior of the system, or equivalently, collective vari-ables because they reflect the macroscopic cooper-ative effect of the enslaved collectives of compo-nents, i.e. synergies. This mechanism provides animmense reduction of information as a conse-quence of the system’s dimension reduction(Haken, 2000, Hristovski et al, 2010). The emergence of such macroscopicallyordered patterns and the information reduction ofbehavior enable us to understand how complexsystems are being coordinated and controlled andhow macroscopic synergies emerge. Here, the term‘synergies’ have to be understood in a generalsense, not only as motor synergies. Psycho-physi-ological synergies (Balague et al, 2012) and teamsynergies (Passos et al, 2008) also show this kindof behavior. It actually solves the problem of infi-nite regress in super-ordinate control devices act-ing in linear control systems discussed in the pre-vious subsection. Order parameters, i.e. collectivevariables, are themselves super-ordinate devicesthat exert control on and coordinate the enslaved,i.e. subordinated components, but this time theyare not pre-formed. They are self-organizeddevices which emerge under the influence of con-trol parameters, i.e. the interacting web of environ-mental, task and organizmic constraints (see e.g.Fox et al, 2005). They are emergent (not pre-struc-tured) and task-environment-organism specificsynergies or functional modules. This mechanismis based on the circular causality in a sense that:subordinate, enslaved, components by their collec-tive, synergic, action form the order parametersand the latter impose the order on them. Eachenslaved component is subject to the collectiveinfluence of all other components, i.e. the collec-tive variable, and is forced to behave cooperative-ly. In this sense we can understand how biologicalas well as sociological substrate self-organizes andengages itself in purposeful functional behaviorunder specific set of constraints that form the con-trol parameter space. These control parameters areusually non-specific. They do not convey the sameinformation as the collective variable. For exam-ple, the perceived distance from the opponent doesnot contain the same information as the actiontoward the opponent. Perceived distance is a per-ceptual variable and the action may be studied as akinematic, kinetic or muscular synergy variable.Hristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 28So, perception is a non-specific control parameterthat does not contain the same information as themovement itself, but it influences i.e. constrains, it.On the other hand a movement action intention isa specific control parameter because it specifies themovement in the same terms of movement param-eters. For example, intention to move on the righthand side contains a specific information of mov-ing on the right hand side. Stable states of collective variables, i.e. orderparameters, are called attractors because theyattract the behavior toward a stable mode of func-tioning (see e.g. Lundqvist et al, 2006). Stabilityitself is defined as a resistance to perturbation. Iffor the same magnitude of perturbation the systemquickly converges to its previous state we say itshows remarkable stability. The attractor is stablebecause it attracts the system quickly. If not, than,we say the system shows tendency to be destabi-lized. The attractor state or dynamics has weakerforcing to pull-in the system. In cases when thesystem qualitatively changes its state we say thatthe system has lost its stability. Lost of stability iswhat happens at the critical points. Close to thesepoints interesting phenomena arise and are gener-al predictions of the nonlinear dynamics theory.One of them has been already described. As con-trol parameters change the system becomes lessstable and more time is needed to recover from theperturbed state. This is called a critical slowingdown phenomenon. Enslaved components by theorder parameter close to the critical point are not“well enslaved” but show larger degree of inde-pendence. Their cooperativity weakens and theyare not strongly pulled toward the attractor anymore. So, to restore the previous order, the onebefore the perturbation took place, the systemneeds more time. So, the restoration takes moretime. This weakening of their cooperativity and theenlarged restoring time makes the system to fluc-tuate in an enhanced manner. Fluctuations, i.e.irregular oscillations around the attractor, enhancefor the same reasons described previously. Thisenhancement of fluctuations is another hallmark ofthe impending critical point and an abrupt qualita-tive change. Some practical applicationsDetecting the order parameters has importantpractical consequences. Because they are the vari-ables that govern the synergic system components,than they should be studied in each sport disciplineseparately and be used as behavioral variables foreither explicit or implicit learning settings.Defining the task in terms of order parametersenhances the formation of coordinative structuresthat have to be learned. For example, telling to thelearner that s/he has to make a strong impact ofhis/her front fist with some surface posited undervarious angles will immediately form a coordina-tive structure that resembles a hook, uppercut orjab pattern. Thus, the angle of impact acts as a col-lective variable that enslaves all other componentsof the arm, i.e. specific joint angles and their rela-tive accelerations. This happens because the angleof impact is the macroscopic variable, i.e. an orderparameter, which the brain uses to plan this dis-crete action. Instructing the learner explicitly, howeach joint angle of the arm should relate to eachother angle may take much time and lead to infor-mation overload. Because the order parameter con-tains this very information in a reduced way thelearning is much quicker. Studying the constraintsthat give rise to different forms of self-organizedpatterns is another important direction of researchin each sport discipline. This approach emphasizesthe individual way of exploring and finding idio-syncratic task solutions of athletes. Much lessexplicit information would be needed for the learn-er to find her/his own behavioral pattern. Lessexplicit information leads to a more robustacquirement of the perception-action couplingsand less instability in ego-threatening situationssuch as sports contests (Poolton et al, 2007).Generally this approach leads to a more independ-ent decision/action making abilities in athletes.Furthermore, learning and training within welldefined constraint sets optimizes the action versa-tility and coordination switching abilities in ath-letes (Hristovski, 2006, Pinder et al, 2011).Training at the areas of action reorganization, i.e.critical points, would enhance the sensitivity ofdetection of specifying perceptual information forregulating the action. The induction of environ-mental-task noise, i.e. fluctuations, strongly facili-tates the exploration and the emergence of novelsolutions to the task goals, even creating novel taskgoals, which on the long term may prove instru-mental in facilitating the creative athlete-environ-ment relationships (Hristovski et al, 2011). Copingstrategies and attentional focus during hard exer-cise show instabilities akin to those describedHristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 29above. Their success strongly depends on their sta-bility profile. Thus, different specific coping strat-egy training would be required for different phasesof exercise (Balague et al, 2012). Many otherpractical interventions stem from the nonlinearpedagogy approach (see e.g. Chow et al, 2011 andthe references therein). In summary, the view ofnonlinear complex systems approach changes thetheoretical basis, the research concepts and thepractical applications to sport related phenomena.Hristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 30REFERENCESAgrachev, A.A., Morse, A.S., Sontag, E.D., Sussmann, H.J., & Utkin, V.I. (2004). Nonlinear and optimal con-trol theory. Springer, Heidelberg. Ashby, W.R. (1966), Design for a Brain. Chapman & Hall, London.Balagué, N., Hristovski, R., Aragonés, D., & Tenenbaum, G. (2012). Nonlinear model of attention focus duringac-cumulated effort. Psychology of Sport and Exercise, 13, 5, 691-697.Chow, J.Y., Davids, K., Hristovski, R., Araújo, D., Passos, P. (2011). Nonlinear pedagogy: Learning design for self-organizing neurobiological systems. New Ideas in Psychology, 29,2, 189-200. Edelman, G. (1987). Neural Darwinism: The Theory of Neuronal Group Selection. Basic Books, NewYork.Edelman, G. (1999). Building a Picture of the Brain. Annals of the New York Academy of Sciences,882, 1, 68-89.Edelman, G. & Gally, (2001). Degeneracy and complexity in biological systems. Proceedings of theNational Academy of Sciences, 98, 24, 13763–13768.Fingelkurts, A. A., & Fingelkurts, A. A. (2004). Making complexity simpler: multivariability and metastability inthe brain. The International Journal of Neuroscience, 114, 843-862.Fox, M. D., Snyder, A. Z., Vincent, J. L., Corbetta, M., Van Essen, D. C., & Raichle, M. E. (2005). The humanbrain is intrinsically organized into dynamic, anticorrelated functional networks. Proceedings of the National Academy of Sciences, 27, 9673-9678.Freeman,W. J., & Holmes, M. D. (2005). Metastability, instability and state transition in neocortex. NeuralNetworks, 18, 497-504.Friston, K. J. (1997). Transients, metastability, and neuronal dynamics. Neuroimage, 5, 164-171.Haken, H. (2000). Information and Self-Organization. A Macroscopic Approach to Complex Systems. Springer,Heidelberg.Hristovski, R., Davids, K., & Araújo, D. (2006). Affordance-controlled bifurcations of action patterns in martialarts. Nonlinear Dynamics, Psychology, & Life Sciences, 10, 409-444.Hristovski, R., Davids, K., & Araújo, D. (2009). Information for regulating actionin sport: Metastability ande-mergence of tactical solutions under ecological constraints. In D. Araujo, H. Ripoll, & M. Raab (Eds.),Perspectives on cognition and action in sport (pp. 43-57). Hauppauge, NY: Nova Science Publishers.Hristovski, R., Venskaityte, E., Vainoras, A., Balagué, N. Vazquez, P. (2010). Constraints controlled metastabledynamics of exercise-induced psychobiological adaptation. Medicina, 46, 7, 447-53.Hristovski, R., Davids, K., & Araújo, D. Passos, P. (2011). Constraints-induced Emergence of Functional Noveltyin Complex Neurobiological Systems: A Basis for Creativity in Sport. Nonlinear Dynamics, Psychology, andLife Sciences, 15, 2, 175-206.Izhikevich, E., Gally, J. A., & Edelman, G. (2004). Spike-timing dynamics of neuronal groups.Cerebral Cortex, 14,933-944.Kello, C. T., Anderson, G. G., Holden, J. G., & Van Orden, G. C. (2008). The pervasiveness of 1/f scaling in speechreflects the metastable basis of cognition. Cognitive Science: A Multidisciplinary Journal, 32, 1217-1231.Lundqvist, M., Rehn, M., Djurfeldt, M., & Lansner, A. (2006). Attractor dynamics in a modular network model of neocortex. Network: Computation in Neural Systems, 17, 253-276.Newell, K. M. (1986). Constraints on the development of coordination. In M. G. Wade & H. T. A. Whiting (Eds.),Motor development in children. Aspects of coordination and control, (pp. 341-360). Dordrecht, Netherlands:Martinus Nijhoff.Paine, R., & Tani, J. (2005). How Hierarchical Control Self-organizes in Artificial Adaptive Systems.AdaptiveBehavior , 13, 3, 211-225.Passos, P., Araújo, D., Davids, K., Gouveia, L., Milho, J., & Serpa, S. (2008). Information-governing dynamics ofattacker–defender interactions in youth rugby union. Journal of Sports Sciences, 26, 1421–1429.Pinder R, Davids K, Renshaw I. (2012) Metastability and emergent performance of dynamic interceptiveHristovski, R., et. al. : LINEAR AND NONLINEAR COMPLEX PESH 1(2012) 1:25-31 31actions,Journal of Science and Medicine in Sport , 15,1,1-7.Poolton, J.M., Masters, R.S.W., Maxwell, J.P. (2007). Passing thoughts on the evolutionary stability of implicitmotor behaviour: Performance retention under physiological fatigue. Consciousness and Cognition, 16, 2,456–468.PRIMENA NA LINEARNI I NELINEARNI KOMPLEKSI SISITEMIVO SPORTOT. EKSPLANATORNI RAZLIKI I PRIMENAUDK:796.011.1:37.013(Pregledna statija)Robert Hristovski1, Natalia Balage2, Boro Daskalovski1 Vujica @ivkovi}1,Len~e Aleksovska-Veli~kovska1, i Milan Naumovski11 Univerzitet „Sv. Kiril i Metodij” vo Skopje, Fakultet za fizi~ka kultura,Skopje, MakedonijaApstrakt:Davame osvrt na najva`nite eksplanatorni razliki pome|u linearniot i nelin-eatrniot teoretski pristap kon slo`enite dinami~ki sistemi vo sportot. Nakratkose osvrnuvame i na nekoi primeni koi se vo razvoj vo ramkite na nelinearniot pristap. Klu~ni zborovi: sport, linearni i nelinearni dinami~ki sistemi, kompleksni sistemiCorrespondence: Robert HristovSs. Cyril and Methodius University in Skopje Faculty of Physical Culture, Zeleznièka b.b. 1000, Skopje, Macedonia e-mail: robert_hristovski@yahoo.com32View publication statsView publication statshttps://www.researchgate.net/publication/240589732
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