Dynamical systems theory From Wikipedia, the free encyclopedia Jump to: navigation, search Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or di fference equations. When differential equations are employed, the theory is call ed continuous dynamical systems. When difference equations are employed, the the ory is called discrete dynamical systems. When the time variable runs over a set which is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set then one gets dynamic equations on time scales. Some situations may also be modelled by mixed operators such as dif ferential-difference equations. This theory deals with the long-term qualitative behavior of dynamical systems, and the studies of the solutions to the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary orbits as well as the behaviour of electronic circuits and the solutions to partial differ ential equations that arise in biology. Much of modern research is focused on th e study of chaotic systems. This field of study is also called just Dynamical systems, Systems theory or lon ger as Mathematical Dynamical Systems Theory and the Mathematical theory of dyna mical systems.

The Lorenz attractor is an example of a non-linear dynamical system. Studying th is system helped give rise to Chaos theory.

Contents [hide] 1 Overview 2 History 3 Concepts 3.1 Dynamical systems 3.2 Dynamicism 3.3 Nonlinear system 4 Related fields 4.1 Arithmetic dynamics 4.2 Chaos theory 4.3 Complex systems 4.4 Control theory 4.5 Ergodic theory 4.6 Functional analysis 4.7 Graph dynamical systems 4.8 Projected dynamical systems 4.9 Symbolic dynamics 4.10 System dynamics 4.11 Topological dynamics 5 Applications 5.1 In biomechanics

5.2 In cognitive science 5.3 In human development 6 See also 7 Notes 8 Further reading 9 External links [edit] Overview Dynamical systems theory and chaos theory deal with the long-term qualitative be havior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but r ather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the l ong-term behavior of the system depend on its initial condition?" An important goal is to describe the fixed points, or steady states of a given d ynamical system; these are values of the variable which won't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it will converge towards the fixed point. Similarly, one is interested in periodic points, states of the system which repe at themselves after several timesteps. Periodic points can also be attractive. S harkovskii's theorem is an interesting statement about the number of periodic po ints of a one-dimensional discrete dynamical system. Even simple nonlinear dynamical systems often exhibit almost random, completely unpredictable behavior that has been called chaos. The branch of dynamical syste ms which deals with the clean definition and investigation of chaos is called ch aos theory. [edit] History The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution r ule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. Before the advent of fast computing machines, solving a dynamical system require d sophisticated mathematical techniques and could only be accomplished for a sma ll class of dynamical systems. Some excellent presentations of mathematical dynamic system theory include Beltr ami (1987), Luenberger (1979), Padulo and Arbib (1974), and Strogatz (1994).[1] [edit] Concepts [edit] Dynamical systems Main article: Dynamical system (definition) The dynamical system concept is a mathematical formalization for any fixed " which describes the time dependence of a point's position in its ambient . Examples include the mathematical models that describe the swinging of a pendulum, the flow of water in a pipe, and the number of fish each spring lake. "rule space clock in a

A dynamical system has a state determined by a collection of real numbers, or mo

re generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical spacea manifold. The evolution rule of t he dynamical system is a fixed rule that describes what future states follow fro m the current state. The rule is deterministic: for a given time interval only o ne future state follows from the current state. [edit] Dynamicism Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cogn itive science or dynamic cognition, is a new approach in cognitive science exemp lified by the work of philosopher Tim van Gelder. It argues that differential eq uations are more suited to modelling cognition than more traditional computer mo dels. [edit] Nonlinear system Main article: Nonlinear system In mathematics, a nonlinear system is a system which is not linear, i.e. a syste m which does not satisfy the superposition principle. Less technically, a nonlin ear system is any problem where the variable(s) to be solved for cannot be writt en as a linear sum of independent components. A nonhomogeneous[clarification nee ded] system, which is linear apart from the presence of a function of the indepe ndent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed t o a linear system as long as a particular solution is known. [edit] Related fields [edit] Arithmetic dynamics Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two a reas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic propertie s of integer, rational, p-adic, and/or algebraic points under repeated applicati on of a polynomial or rational function. [edit] Chaos theory Chaos theory describes the behavior of certain dynamical systems C that is, syst ems whose state evolves with time C that may exhibit dynamics that are highly sen sitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth o f perturbations in the initial conditions, the behavior of chaotic systems appea rs to be random. This happens even though these systems are deterministic, meani ng that their future dynamics are fully defined by their initial conditions, wit h no random elements involved. This behavior is known as deterministic chaos, or simply chaos. [edit] Complex systems Complex systems is a scientific field, which studies the common properties of s ystems considered complex in nature, society and science. It is also called comp lex systems theory, complexity science, study of complex systems and/or sciences of complexity. The key problems of such systems are difficulties with their for mal modeling and simulation. From such perspective, in different research contex ts complex systems are defined on the base of their different attributes. The st udy of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefo re often used as a broad term encompassing a research approach to problems in ma ny diverse disciplines including neurosciences, social sciences, meteorology, ch emistry, physics, computer science, psychology, artificial life, evolutionary co mputation, economics, earthquake prediction, molecular biology and inquiries int

o the nature of living cells themselves. [edit] Control theory Control theory is an interdisciplinary branch of engineering and mathematics, t hat deals with influencing the behavior of dynamical systems. [edit] Ergodic theory Ergodic theory is a branch of mathematics that studies dynamical systems with a n invariant measure and related problems. Its initial development was motivated by problems of statistical physics. [edit] Functional analysis Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It ha s its historical roots in the study of functional spaces, in particular transfor mations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function . Its use in general has been attributed to mathematician and physicist Vito Vol terra and its founding is largely attributed to mathematician Stefan Banach. [edit] Graph dynamical systems The concept of graph dynamical systems (GDS) can be used to capture a wide rang e of processes taking place on graphs or networks. A major theme in the mathemat ical and computational analysis of GDS is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result. [edit] Projected dynamical systems Projected dynamical systems is a mathematical theory investigating the behaviou r of dynamical systems where solutions are restricted to a constraint set. The d iscipline shares connections to and applications with both the static world of o ptimization and equilibrium problems and the dynamical world of ordinary differe ntial equations. A projected dynamical system is given by the flow to the projec ted differential equation. [edit] Symbolic dynamics Symbolic dynamics is the practice of modelling a topological or smooth dynamica l system by a discrete space consisting of infinite sequences of abstract symbol s, each of which corresponds to a state of the system, with the dynamics (evolut ion) given by the shift operator. [edit] System dynamics System dynamics is an approach to understanding the behaviour of complex system s over time. It deals with internal feedback loops and time delays that affect t he behaviour of the entire system.[2] What makes using system dynamics different from other approaches to studying complex systems is the use of feedback loops and stocks and flows. These elements help describe how even seemingly simple sys tems display baffling nonlinearity. [edit] Topological dynamics Topological dynamics is a branch of the theory of dynamical systems in which qu alitative, asymptotic properties of dynamical systems are studied from the viewp oint of general topology. [edit] Applications [edit] In biomechanics In sports biomechanics, dynamical systems theory has emerged in the movement sci ences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscula r, perceptual) that are composed of a large number of interacting components (e. g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective t issue and bone). In dynamical systems theory, movement patterns emerge through g eneric processes of self-organization found in physical and biological systems.[ 3] [edit] In cognitive science

Dynamical system theory has been applied in the field of neuroscience and cognit ive development, especially in the neo-Piagetian theories of cognitive developme nt. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI. It also believed that diff erential equations are the most appropriate tool for modeling human behavior. Th ese equations are interpreted to represent an agent's cognitive trajectory throu gh state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted. In it, the learner's mind reaches a state of disequilibrium where old patterns h ave broken down. This is the phase transition of cognitive development. Self-org anization (the spontaneous creation of coherent forms) sets in as activity level s link to each other. Newly formed macroscopic and microscopic structures suppor t each other, speeding up the process. These links form the structure of a new s tate of order in the mind through a process called scalloping (the repeated buil ding up and collapsing of complex performance.) This new, novel state is progres sive, discrete, idiosyncratic and unpredictable.[4] Dynamic systems theory has recently been used to explain a long-unanswered probl em in child development referred to as the A-not-B error.[5] [edit] In human development Dynamic systems theory is a psychological theory of human development. Unlike dy namical systems theory which is a mathematical construct, dynamic systems theory is primarily non-mathematical and driven by qualitative theoretical proposition s. This psychological theory does, however, apply metaphors derived from the mat hematical concepts of dynamical systems theory to attempt to explain the existen ce of apparently complex phenomena in human psychological and motor development. As it applies to developmental psychology, this psychological theory was develop ed by Esther Thelen, Ph.D. at Indiana University-Bloomington. Thelen became inte rested in developmental psychology through her interest and training in behavior al biology. She wondered if "fixed action patterns," or highly repeatable moveme nts seen in birds and other animals, were also relevant to the control and devel opment of human infants [6] According to Miller (2002),[7] dynamic systems theory is the broadest and most e ncompassing of all the developmental theories??????. ??????This theory attempts to encompass all the possible factors that may be in operation at any given deve lopmental moment; it considers development from many levels (from molecular to c ultural) and time scales (from milliseconds to years).[7] Development is viewed as constant, fluid, emergent or non-linear, and multidetermined.[8] Dynamic syst ems theorys greatest impact has been in early sensorimotor development.[6] Esther Thelen believed that development involved a deeply embedded and continuou sly coupled dynamic system. It is unclear however if her utilization of the conc ept of "dynamic" refers to the conventional dynamics of classical mechanics or t o the metaphorical representation of "something that is dynamic" as applied in t he colloquial sense in common speech, or both. The typical view presented by R.D . Beer showed that information from the world was given to the nervous system wh ich directs the body, which in turn interacts back on the world. Esther Thelen i nstead offers a developmental system that has continual and bidirectional intera ction between the world, nervous system and body. The exact mechanisms for such interaction, however, remain unspecified.[9] The dynamic systems view of development has three critical features that separat

e it from the traditional input-output model. The system must first be multiply causal and self-organizing. This means that behavior is a pattern formed from mu ltiple components in cooperation with none being more privileged than another. T he relationship between the multiple parts is what helps provide order and patte rn to the system. Why this relation would provide such order and pattern, howeve r, is unclear. Second, a dynamic system is a dependent on time making the curren t state a function of the previous state and the future state a function of the current state. The third feature is the relative stability of a dynamic system. For a system to change, a loose stability is needed to allow for the components to reorganize into a different expressed behavior. What constitutes a stability as being loose or not-loose, however, is not specified. Parameters that dictate what constitutes one state of organization versus another state are also not spe cified, as a generality, in dynamic systems theory. The theory contends that dev elopment is a sequence of times where stability is low allowing for new developm ent and where stability is stable with less pattern change. The theory contends that in order to make these movements, you must scale up on a control parameter in order to reach a threshold (past a point of stability). Once that threshold i s reached, the muscles will begin to form the different movements. This threshol d must be reached in order for each different muscle to contract and relax to ma ke the movement. The theory can be seen to present a variant explanation for mus cle length-tension regulation but the extrapolation of a vaguely outlined argume nt for muscle action to a grand theory of human development remains unconvincing and unvalidated.[10] Esther Thelen's early research in infant motor behavior (particularly stepping, kicking, and reaching) led her to become dissatisfied with existing theories and moved her toward a dynamic systems perspective. Prior views of development conc eptualized infants as passive and infants motor development as the result of a gen etically determined developmental plan. Thelen, in her work, contended that infa nts' body weights and proportions, postures, elastic, and inertial properties of muscle and the nature of the task and environment contribute equally to the mot or outcome. None of these contentions have been scientifically validated due in part to the breadth and poor operational definition of the parameters used to re present the phenomena involved. It is theorized that infants can "self-assemble" new motor patterns in novel situations, but what this actually means awaits fur ther and specific clarification. The theory contends that development happens in individual children solving individual problems in their own unique ways.[8] Th elen used the proposition that because each child is different in terms of his o r her body, nervous system, and daily experience, the course of development is n early impossible to predict, and yet the theory does not account for clear trend s and predictability in development for most children, despite there being multi ple pathways to development.[11] Development is supposedly not just the result o f genetics or the environment, but rather the interweaving of events at a given moment.[11] How such interweaving occurs is not specified by the theory in certa in terms. Dynamic systems theory proponents claim to have had the greatest impac t on early sensorimotor development.[6]??????? [edit] See also Related subjects List of dynamical system topics Baker's map Biological applications of bifurcation theory Dynamical system (definition) Embodied Embedded Cognition Gingerbreadman map Halo orbit List of types of systems theory Oscillation Postcognitivism Recurrent neural network Combinatorics and dynamical systems

Synergetics Systemography Related scientists People in systems and control Dmitri Anosov Vladimir Arnold Nikolay Bogolyubov Andrey Kolmogorov Nikolay Krylov Jrgen Moser Yakov G. Sinai Stephen Smale Hillel Frstenberg Grigory Margulis Elon Lindenstrauss Esther Thelen [edit] Notes 1.^ Jerome R. Busemeyer (2008), "Dynamic Systems". To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. 2.^ MIT System Dynamics in Education Project (SDEP) 3.^ Paul S Glaziera, Keith Davidsb, Roger M Bartlettc (2003). "DYNAMICAL SYSTEM S THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Rese arch". in: Sportscience 7. Accessdate=2008-05-08. 4.^ Lewis, Mark D. (2000-02-25). "The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development" (PDF). Child Development 71 (1): 36C 43. doi:10.1111/1467-8624.00116. PMID 10836556. Retrieved 2008-04-04. 5.^ Smith, Linda B.; Esther Thelen (2003-07-30). "Development as a dynamic syst em" (PDF). TRENDS in Cognitive Sciences 7 (8): 343C8. doi:10.1016/S1364-6613(03)0 0156-6. Retrieved 2008-04-04. 6.^ a b c Template:Thelen, E., & Bates, E. (2003). Connectionism and dynamic sy stems: Are they really different? Developmental Science, 6, 378-391. 7.^ a b Template:Miller, P. (2002). Theories of Developmental Psychology (4th e d.). New York, NY: Worth Publishers. 8.^ a b Template:Spencer, J. P., Clearfield, M., Corbetta, D., Ulrich, B., Buch anan, P., & Sch?ner, G. (2006). Moving toward a grand theory of development: In memory of Esther Thelen. Child Development, 77, 1521-1538. 9.^ Thelen, E. (2000), Grounded in the World: Developmental Origins of the Embo died Mind. Infancy, 1: 3C28. doi:10.1207/S15327078IN0101_02 10.^ Thelen, E. (2000), Grounded in the World.